SCHAUM'S
MATHEMATICAL
HANDBOOK
off FORMULAS and TABLES
by MURRAY R. SPIEGEL
including
fformulcis
I^O tables
SCHAUM'S OUTLINE SERIES
McGRAW-HILL BOOK COMPANY
^n^m
SCHAUM'S OVTLmE SERIES
MATHEMATICAL
HANDBOOK
of
Formulas and Tables
BY
MURRAY R. SPIEGEL, Ph.D.
Professor of Mathematics
Rensselaer Polytechnic Institute
SCHAUM'iS OIJTLIAE SERIES
McGRAW-HILL BOOK COMPANY
New York, St. Louis, ^um^rancis£a»Soronlo, Sydney
.- 'f ■ ^
Copyright © 1968 by McGraw-Hill. Inc. All Rights Reserved. Printed in the
United States of America. No part of this publication may be reproduced,
stored in a retrieval system, or transmitted, in any form or by any means,
electronic, mechanical, photocopying, recording, or otherwise, without the
prior written permission of the publisher.
60224
234567890 MHUN 72 10 698
Tvpoaravhv by Signs and Symbola. Inc., New York, N. Y.
Preface
The purpose of this handbook is to supply a collection of mathematical formulas and
tables which will prove to be valuable to students and research workers in the fields of
mathematics, physics, engineering and other sciences. To accomplish this, care has been
taken to mclude those formulas and tables which are most likely to be needed in practice
rather than highly specialized results which are rarely used. Every effort has been made
to present results concisely as well as precisely so that they may be referred to with a maxi-
mum of ease as well as confidence.
Topics covered range from elementary to advanced. Elementary topics include those
from algebra, geometry, trigonometry, analytic geometry and calculus. Advanced topics
mclude those from differential equations, vector analysis, Fourier series, gamma and beta
functions, Bessel and Legendre functions, Fourier and Laplace transforms, elliptic functions
and various other special functions of importance. This wide coverage of topics has been
adopted so as to provide within a single volume most of the important mathematical results
needed by the student or research worker regardless of his particular field of interest or
level of attainment.
The book is divided into two main parts. Part I presents mathematical formulas
together with other material, such as definitions, theorems, graphs, diagrams, etc., essential
for proper understanding and application of the formulas. Included in this first part are
extensive tables of integrals and Laplace transforms which should be extremely useful to
the student and research worker. Part II presents numerical tables such as the values of
elementary functions (trigonometric, logarithmic, exponential, hyperbolic, etc.) as well as
advanced functions (Bessel, Legendre, elliptic, etc.). In order to eliminate confusion
especially to the beginner in mathematics, the numerical tables for each function are sep-
arated. Thus, for example, the sine and cosine functions for angles in degrees and minutes
are given in separate tables rather than in one table so that there is no need to be concerned
about the possibility of error due to looking in the wrong column or row.
I wish to thank the various authors and publishers who gave me permission to adapt
data from their books for use in several tables of this handbook. Appropriate references
to such sources are given next to the corresponding tables. In particular I am indebted to
the Literary Executor of the late Sir Ronald A. Fisher, F.R.S., to Dr. Frank Yates F R S
and to Oliver and Boyd Ltd., Edinburgh, for permission to use data from Table III of their
book Statistical Tables for Biological, AgHcultural and Medical Research.
I also wish to express my gratitude to Nicola Monti, Henry Hayden and Jack Margolin
for their excellent editorial cooperation.
M. R. Spiegel
Rensselaer Polytechnic Institute
September, 1968
CONTENTS
Page
1 . Special Constants 1
2. Special Products and Factors 2
3. The Binomial Formula and Binomial Coefficients 3
4. Geometric Formulas 5
5. Trigonometric Functions ^1
6. Complex Numbers 21
7. Exponential and Logarithmic Functions 23
8. Hyperbolic Functions 26
9. Solutions of Algebraic Equations 32
10. Formulas from Plane Analytic Geometry 34
11. Special Plane Curves 40
12. Formulas from Solid Analytic Geometry 46
13. Derivatives ^
14. Indefinite Integrals 57
15. Definite Integrals ^
16. The Gamma Function 101
17. The Beta Function 103
1 8. Basic Differential Equations and Solutions 104
19. Series of Constants 107
20. Taylor Series HO
21 . Bernoulli and Kuler Numbers 114
22. Formulas from Vector Analysis H®
23. Fourier Series 131
24. Bessel Functions 13^
25. Legendre Functions 14®
26. Associated Legendre Functions 149
27. Hermite Polynomials 1^1
28. Laguerre Polynomials 1^
29. Associated Laguerre Polynomials 1^5
30. Chebyshev Polynomials 1^*^
CONTENTS
Page
31. Hypergeometric Functions 160
32. Laplace Transforms 161
33. Fourier Transforms 174
34. Elliptic Functions 179
35. Miscellaneous Special Functions 183
36. Inequalities 185
37. Partial Fraction Expansions 187
38. Infinite Products 188
39. Probability Distributions 189
40. Special Moments of Inertia 190
41. Conversion Factors 192
Sample problems illustrating use of the tables 194
1. Four Place Common Logarithms 202
2. Four Place Common Antilogarithms 204
3. Sin X (x in degrees and minutes) 206
4. Cos X {x in degrees and minutes) 207
5. Tan x (x in degrees and minutes) 208
6. Cotx {x in degrees and minutes) 209
7. Sec X {x in degrees and minutes) 210
8. Csc X {x in degrees and minutes) 211
9. Natural Trigonometric Functions (in radians) 212
10. log sin X {x in degrees and minutes) 216
11. log cos X {x in degrees and minutes) 218
12. log tan X {x in degrees and minutes) 220
13. Conversion of radians to degrees, minutes and seconds
or fractions of a degree 222
14. Conversion of degrees, minutes and seconds to radians 223
15. Natural or Napierian Logarithms lege a; or In a; 224
16. Exponential functions c* 226
17. Exponential functions e"'' 227
18a. Hyperbolic functions sinh x 228
18b. Hyperbolic functions cosh x 230
18e. Hyperbolic functions tanhx 232
CONTENTS
Page
19. Factorial n 234
20. Gamma Function 234
21. Binomial Coefficients 236
22. Squares, Cubes, Roots and Reciprocals 238
23. Compound Amount: (1 + r)" 240
24. Present Value of an Amount: (1 + r)"" 241
25. Amount of an Annuity: (^+^)"'~ ^ 242
r
26. Present Value of an Annuity: ^^'^'^^
r 243
27. Bessel functions Jo{x) 244
28. Bessel functions Ji{x) 244
29. Bessel functions Yq{x) 245
30. Bessel functions Yi{x) 245
31. Bessel functions Iq{x) 246
32. Bessel functions Ii{x) 246
33. Bessel functions Ko{x) 247
34. Bessel functions Ki{x) 247
35. Bessel functions Ber (x) 248
36. Bessel functions Bei (x) 248
37. Bessel functions Ker (x) 249
38. Bessel functions Kei (x) 249
39. Values for Approximate Zeros of Bessel Functions 250
40. Exponential, Sine and Cosine Integrals 251
41. Legendre Polynomials Pn{x) 252
42. Legendre Polynomials Pr.{cos 6) 253
43. Complete Elliptic Integrals of First and Second Kinds 254
44. Incomplete Elliptic Integral of the First Kind 255
45. Incomplete Elliptic Integral of the Second Kind 255
46. Ordinates of the Standard Normal Curve 256
47. Areas under the Standard Normal Curve 257
48. Percentile Values for Student's t Distribution 258
49. Percentile Values for the Chi Square Distribution 259
50. 95th Percentile Values for the F Distribution 260
51. 99th Percentile Values for the F Distribution 261
52. Random Numbers 262
Index of Special Symbols and Notations 263
Index 265
Part I
FORMULAS
THE GREEK ALPHABET
1
Greek
name
Greek letter
Lower case
Capital
Alpha
CC
A
Beta
/3
B
Gamma
y
r
Delta
s
A
Epsilon
£
E
Zeta
t
Z
Eta
V
H
Theta
9
Iota
I
I
Kappa
K
K
Lambda
A
A
Mu
t^
M
Greek
name
Greek letter
Lower case
Capit£.l
Nu
V
N
Xi
1
H
Omicron
Pi
TT
n
Rho
P
P
SigTTia
a
2
Tau
T
T
Upsilon
V
Y
Phi
*
*
Chi
X
X
Psi
^
*
Omega
u
a
1.1 TT = 3.14159 26535 89793 23846 2643...
1.2 e = 2.71828 18284 59045 23536 0287... = lim f 1 + -)"
= natural base of logarithms «-» \ n/
1.3 V2 =: 1.4142135623 73095 0488...
1.4 ^/3 := 1.73205 08075 68877 2935...
IS v^ ^ 2.23606 79774 99789 6964...
3 _
1.6 V2 = 1.25992 1050...
1.7 ^/S = 1.44224 9570...
1.8 y/2 = 1.14869 8355...
1.9 ^ = 1.24573 0940...
1.10 e"" = 23.14069 26327 79269 006...
1.11 n-e = 22.45915 77183 61045 47342 715...
1.12 e^ = 15.15426 22414 79264 190...
1.13 log,o2 = 0.30102 99956 639811952137389...
1.14 logioS ^ 0.47712 12547 19662 43729 50279...
1-15 logio e = 0.43429 44819 0325182765...
1.16 log,o,r = 0.49714 98726 94133 85435 12683...
1.17 log^ 10 ^ InlO = 2.30258 50929 94045 684017991...
1.18 ]og^2 = ln2 = 0.69314 71805 59946 309417232...
1.19 log, 3 = In 3 = 1.0986122886 68109 69139 5245...
y = 0.5772156649 01532 86060 6512... = Euler's constant
1.23
1.24
1.25
1.20
1.21 ey ^ 1.78107 24179 90197 9852... [see 1.20]
1.22 ^fe = 1.64872 12707 00128 1468...
V^ = r(^) = 1.77245 38509 05516 02729 8167...
where r is the gamma function [see pages 101-1021.
r(^) = 2.67893 85347 07748...
r(^} = 3.62560 99082 21908...
1.26 1 radian = l%0°h ~ 57.29577 95130 8232...°
1.27 1° ^ ,r/180 radians ^ 0.01745 32925 19943 2957. .. radians
2.1 ix + y)^ = x2 + 2xy + y^
2.2 {X - j/)2 = x2 - 2xy + y^
2.3 (a: + j/)3 = x3 + 3x2]/ + Sxy2 + j,3
2.4 (x - ^)3 = x3 - 3a:2i/ + 3x1/2 - j/3
2.5 (X + y)* = X* + iX^y + 6x2j/2 + 4a;y3 + j^
2.6 (» - y)-* = X*- Ax^y + 6x2]/2 - 4xy3 + y^
2.7 (X + y)5 = x5 + 5r»j/ + 10x3i/2 + 10x2y3 + ^y;yi + j^5
2.8 {x - j/)5 = x5 - 5x^1/ + 10x3^2 _ I0x2y3 + 5a:y4 _ yS
2.9 (x + y)^ = x« + 6x5i/ + 15x4i/2 + 20xV + I5a:2y4 + gxyS + y6
2.10 (x - i/)fl = x8 - 6x5y + 15»*i/2 - 20x3tf3 + 15x2y4 - 6x^/5 + y^
The results 2.1 to 2.10 above are special cases of the binomial formula [see page 3].
2.11 a:2- y2 = (a; - y)(a; + y)
2.12 x3-|/3 = (X - 1/)(X2 + xy + 1/2)
2.13 X3 + y3 = (;;e + y)(a;2_a;y + y2)
2.14 z*~y* = (a: - y)(x + y)(x2 4- y2)
2.15 x5 - yS = (x-y)(x4 + a;3y + x2y2 + a:y3 + y4)
2.16 x5 + y5 = (x + y)(x* — x3y + x2y2 - xy3 + y*)
2.17 x6-y6 = {x-y)ix + y)(x^ + xy-hy^){x^-xy + yi)
2.18 X4 + X2y2 + y4 = (a;2 + :cy + y2)(a;2 _ a;y + y2)
2.19 x* + 43/4 = (a.2 + 2xy+2y2)(x2-2xj/ + 2y2)
Some generalizations of the above are given by the following results where m is a positive integer.
2.20 x2" + l - y2n + i = (a- - y)(a;2" + x2n-ly + x2"-2y2 + . - . + y2n)
2xy cos
= (x - y) X
x2 — 2xy cos
2n + 1
2nir
2n+l
+ y2 )( sc2 — 2xy cos
+ y2
Air
2n + l
+ y2
2.21 x2"+l + y2n+l = (x + y)(x2n - x2''-ly + x2n-2y2 _ . . . + y2n)
2
= (x + y) { x2 + 2xy cos^ — x~r + y^ )( a^ + 2xy cos _
x2 + 2xy cos ^ — -^ + y2
^.vO
2.22
r2n _ ,,2n —
— (* ~ y)(3^ + y)(a:"~' + x'*~2y + 3.n-3y2 + . . .)(a.n-i _ x^^^y + a;"~3y:i _ . . .)
— (sJ ~ J/){^ 4- y) ( x2 — 2xy cos— + y2 )[ x2 — 2xy eos-^ + y^
■ ~ 2xy cos ^" ~ ^^"' + y2
2^3
+ y2n - ( x2 + 2xy cos^ + y2 J( a;2 + 2xy cos |^ + y
2n
2n
•} I n (2n — l)jr , n
x'- + 2a:y cos -z + y2
3
The BINOMIAL FORMULA
and BINOMIAL COEFFICIENTS
If 71 — 1,2,3, . . . factorial n or n factorial is defined as
3.1 n! = 1-2-3 n
We also define zero factorial as
3.2
0! ^ 1
mwf^m fm^im mB^
If 71 = 1,2,3, ... then
3.3 {x + vr = a;" + nx^^^y + ^^^^^f^ a:"-2y2 + "^"^ ~ g^"" ^^ a:"^3y3 + ... + y«
This is called the binomta? formula. It can be extended to other values of n and then is an infinite series
[see Binomial Series, page 110].
BINOMIAL COEFFICIENTS
imiiiiiiiiiiiiiiii^
The reaxilt 3.3 can also be written
3.4 (x + j/)" = x^ + ('^Jx'^-^y + f 2Jx"-2y2 + [^ )x^-3y^ +
where the coeflicients, called binomial coefficients, are given by
3.5
n\ ^ n{n - l){n - 2)- • -{n- k + 1) ^ n\
kj k\ k\(n-k)\
'n
n
n ~ k
4
THE BINOMIAL FORMULA AND BINOMIAL COEFFICIENTS
PROPERTIES OF BINOMIAL COEFFICIENTS
3.6
+
« + 1
k-{- 1
k / V A; + 1
This leads to Paacal's triangle [see page 236].
3.7
n\ in
+
ly + UJ +
+
= 2"
3.8
(;)-(:)-(;)-•■•<-(:)
=
3.9
C)
■nl \ n I \ n I \ n
« + »i + 1
n + 1
3.10
C) - (:
+ (4 ) +
= %v.-\
3.11
vm:^-^"'-
= 2"-*
3.12
+
+
+ ■■■ +
3.13
:)(:)-(:)(.-.-
m + m
P
3.14
/n
(Di^lJ + (2)1^2; + (3}(^3J + ■
+ («1
■= n2"-'
/?i
3.15 (l)(^i^ - (2)(^2y + (3)(^3
,(„!)« + !(„)( ^ ) =
MULTINOMIAL FORMULA
3.16
(ici + a:^ + ■ • ■ + iCp)" = 5
w! _n, „n« _n,
where the sum, denoted by 5, is taken over all nonnegative integers n^tU-i,
. , n^ for which
GEOMETRIC FORMULAS
RECTANGLE OF LENGTH 6 AND WIDTH o
4.1 Area = ab
4.2 Perimeter = 2a + 2b
b
Fig. 4-1
PARALLELOGRAM OF ALTITUDE h AND BASE h
4.3 Area = bk = ab sin e
4.4 Perimeter = 2a + 26
b
Fig. 4-2
4.5
RIANGLE OF ALTITUDE h AND BASE 5
Area = ^hh = ^ab sin e
where s^ ^{a + b + c) = semiperimeter
4.6 Perimeter = a + b + c
4.7
4.8
TRAPEZOID OP ALTITUDE h AND PARALLEL SIDES a A
N
Area - ^k{a + b)
Perimeter = a + b -i- h ( — L -\ ^
\sinff sin^
= a + b + /i(csc e + CSC -p)
b
Pig. 4-4
GEOMETRIC FORMULAS
4.9
4.10
4.15
REGULAR POLYGON OF n SIDES EACH OF LENGTH b
Area = ^nb^ cot— = Inb^^
cos JTr/n)
sin (ir/n)
Perimeter = nb
CIRCLE OF RADIUS r
4.11 Area = trr^
4.12 Perimeter = 2irr
SECTOR OF CIRCLE OF RADIUS r
4.13 Area - ]^r^$ [ff in radians]
4.14 Arc length 8 = re
RADIUS OF CIRCLE INSCRIBED IN A TRIANG
\/8(« — a)(s — 6)(« — c)
where s = ^(a + b + c) — semiperimeter
Fig. 4-5
Fig. 4-6
RADIUS OF CIRCLE CIRCUMSCRIBING A TRIANGLE OF SIDES a,b,c
4.16
R =
abc
4^/s{8 — a)(s — 6)(s — c)
where s = ^(a+ b + e) — semiperimeter
Fig. 4-9
GEOMETRIC FORMULAS
REGULAR POLYGON OF n SIDES INSCRIBED IN CIRCLE OF RADIUS r
4.17 Area = ^nr^ sin ~ = ^nr^ sin ^^
4.18 Perimeter = 2nrsin— = 2nr sin
n ft
n SIDES CIRCUi
REGULAR POLYG
4.19 Area = nr^ tan- = nr^ t&n^^
n n
4.20 Perimeter = 2nr tan — = 2nr tan ^^
71 n
IE6MENT OF CIRCLE OF RADIUS r
4.21 Area of shaded part = ^+-2 {$ — sin tf)
Fig. 4-11
Fig. 4-12
ELLIPSE
nkmmmMMimkimmm^
4.22 Area = vob
r/2
y/l —
k^ sin2 $ de
— 2v y/^{a^ + b^) [approximately]
where k = Va^ — by a. See page 254 for numerical tables.
fMililiJB^ AXIS b
Fig. 4-13
lEGMENT OF A PARABO
4.24 Area = %ab
, 1,2 /4a + y/b^'+lG^
4.25 Arc length ABC = A VP+Jg^ + ~ In
Fig. 4-14
8
GEOMETRIC FORMULAS
RECTANGULAR PARALLELEPIPED OF LENGTH a, HEIGHT I, WIDTH
4.26 Volume = ahc
4.27 Surface area = 2(o6 + tie + 6c)
a
Fig. 4-15
PARALLELEPIPED OF CROSS-SECTIONAL AREA ^ AND HEIGHT h
4.28 Volume = Ah — ahc si
sin e
Fig. 4-16
SPHERE OF RADIUS r
4.29 Volume := ^.^
4.30 Surface area = Attt'^
Fig. 4-17
RIGHT CIRCULAR CYLINDER OF RADIUS r AND HEIGHT h
4.31 Volume = irr^h
4.32 Lateral surface area = 'i.Trrh
Fig. 4-18
CIRCULAR CYLINDER OF RADIUS r AND SLANT HEIGHT I
4.33 Volume ~ tttH = —. — ■ = trr'^h esc B
sin e
4.34 Lateral surface area = %^tI — ~- — = 2?rr^ esc B
sin B
Fig. 4-19
GEOMETRIC FORMULAS
CYLINDER OF CROSS-SECTIONAl AREA A AND SLANT HEIGHT I
4.35 Volume = Al = — — = Ah esc e
sine
4.36 Lateral surface area = vl = -?■ — ■= ■ph cse B
sme
Note that formulas 4.31 to 4.34 are special cases.
Fig. 4-20
RIGHT CIRCULAR CONE OF RADIUS r AND HEIGHT h
4.37 Volume = lir-r^h
4.38 Lateral surface area - vr v'r2"4~P = wrl
Fig. 4-21
PYRAMID OF BASE AREA A AND HEIGHT h
4.39 Volume = ^Ah
SPHERICAL CAP OF RADIUS r AND HEIGHT h
Fig. 4-22
4.40 Volume {shaded in figure) = ^7r/t2{3r - h)
4.41 Surface area = 2irrh
Fig. 4-23
FRUSTRUM OF RiGHT CIRCULAR CONE OF RADII a.b AND HEIGHT h
4.42 Volume ^ ^^h{a^ -\- ah -\- h^)
4.43 Lateral surface area = Tr{a + h) yjh^ + (6 - a)2
Fig. 4-24
10
GEOMETRIC FORMULAS
PHERICAL TRIANGLE OF ANGLES A,B,C ON SPHERE OF RADIUS r
4.44 Area of triangle ABC = {A + B + C - jr)r'^
Fig. 4-25
TORUS OF INNER RADIUS a AND OUTER RADIUS b
4.45 Volume - iTrHa + b)(b - a)2
4.46 Surface area = Jr2(b2 - ^2)
Pig. 4-26
;OID OF SEMI-AXES a.b^c
4.47 Volume — jvabc
Fig. 4-27
PARABOLOID OF REVOLUTION
4.48 Volume = j^irb^a
Fig. 4-28
DEFINITION O F TRIGONOMETRIC FUNCTIONS FOR A RIGHT TRIANGLE
J
Triangle ABC has a right angle (90°) at C and sides of length a, 6, c. The trigonometric functions of
angle A are defined as follows.
5.1
S.2
5.3
5.4
BS
5.6
. . a, opposite
»i7te of A = sin A = - = .„„„■„„.. —
c hypotenuse
. . A ^ adjacent
coaiTie ot A = cos A = — —
c hypotenuse
, . ^ . O' opposite
tangent ot A - tan A - ^ - ^j^^^
cotangent of A = cot A
& _ adjacent
a opposite
, . . c hypotenuse
secant of A = sec A = -r = — j-- t—
b adjacent
_. . . A ^ hypotenuse
cosecant of A = esc A = — = - — ~ — tt —
a opposite
Fig. 5-1
EXTENSIONS TO ANGLES WHICH MAY BE GREATER THAN 90'
Consider an xy coordinate system [see Fig. 5-2 and 5-3 below]. A point P in the xy plane has coordinates
{x,y) where x is considered as positive along OX and negative along OX' while y is positive along py and
negative along OY'. The distance from origin O to point P is positive and denoted by r — y/x^ + y^ .
The angle A described counterclockwise from OX is considered positive. If it is described clockwise from
OX it is considered negative. We call X'OX and Y'OY the x and y axis respectively.
The various quadrants are denoted by I, II, III and IV called the first, second, third and fourth quad-
rants respectively. In Fig. 5-2, for example, angle A is in the second quadrant while in Fig. 5-3 angle A
is in the third quadrant.
11
12
TRIGONOMETRIC FUNCTIONS
For an angle A in any quadrant the trigonometric functions of A are defined as follows.
5-/ sin A = y/r
5.8
5.9
5.10
5.11
5.12
cos A = xfr
tan A = y/x
cot A = x/y
sec A = r/x
CSC A = r/y
RELATIONSHIP BETWEEN DEGREES AND RADIANS
A radian is that angle 6 subtended at center O of a circle by an arc
MN equal to the radius r.
Since 27r radians = 360° we have
5.13 1 radian ^ 180°/;r = 57.f9577 95130 8232. .. °
5.14 1° = B-/180 radians ^ 0.1745 32925 19943 2957. . .radians
Fig. 5-4
5.15
5.18
RELATIONSHIPS AMONG TRIGONOMETRIC FUNCTIONS
tan A
sin A
5.16 cotA =
5.17 sec A =
CSC A = — :
cos A
1
tan A
cos A
sin A
1
cos A
1
sin A
5.19 sin2A + cos^A = 1
5.20 sec2A - Un2A = 1
5.21 csc2A - cot2A - 1
"AMErVARIATIONS OF TRIGONOMETRIC FUNCTIONS
Quadrant
sin A
cos A
tan A
cot A
sec A
esc A
I
-T
to 1
+
1 to
+
to so
+
=° to
+
1 to «;
+
00 to 1
II
+
1 to
to -1
~« to
to -«
-K to -1
+
1 to =o
III
to -1
-1 too
to ~
-t-
« to
-1 to -»
— re to —1
IV
-I to
+
Otol
-« to
to -«>
+
== to 1
— 1 to —00
TRIGONOMETRIC FUNCTIONS
13
EXACT VALUES FOR TRIGONOMETRIC FUNCTIONS OF VARIOUS ANGLES
Angle A
in degrees
Angle A
in radians
sin A
cos A
tan A
cot A
sec A
CSC A
0°
1
00
1
00
15°
7r/12
liV6-V2)
i(\/6 + V^)
2-^/z
2 + ^/3
^/6-^/2
Ve + \/2
30°
:r/6
i
iVs
iV3
V3
fV3
2
45°
;r/4
IV2
iV^
1
1
\/^
V2
60°
HZ
iVs
1
2
V3
^\/3
2
iV§
75°
5W12
liV6 + V2)
1(Vg-V2)
2 + V^
2-\/3
\/6 + \/2
V6-\/2
90°
7r/2
1
±00
±co
1
105°
77r/12
l{\/6 + \/2)
-i(V6-V^)
-(2 + -/3 )
-{2-V^)
-(\/6 + V2)
\/6- VZ
120°
2B-/3
1^3
-i
-\/3
-^Vs
-2
fv^
135°
3W4
iV2
-iV2
-1
-1
-^
\/^
150°
57r/6
1
-iV3
-*V3
-v^
-fv^
2
165°
lln-/12
1(^6 -V^)
-i(v/6 + \/2)
H2-\/3)
-(2 + V3)
-(\/6-V2)
V6 + \/2
180°
IT
-1
^oo
-1
±00
195°
13W12
~1(Vg-V2)
-i(V6 + \/2)
2- \/3
2 + \/3
-(V6-\/2)
-i^/G + ^/2)
210°
lir/e
-i
-1^3
4V3
V3
-|V3
-2
225°
5)r/4
-^V2
-i\/2
1
1
~V2
-V2
240°
47r/3
-iVa
—1
2
v/3
i\/3
-2
-fV3
255°
1777-/12
-i(V6 + v^)
-HVe~V2)
2 + \/3
2 - VS
-(\/6 + \/2)
-(V6-V2)
270°
37r/2
-1
±0=
^:oo
~1
285°
197r/12
-i(V6 + \/2)
i{V6--v/2)
-(2 + \/3 )
~(2~^/Z)
V^ + \/2
-(\/6-\/2)
300°
577/3
-iVs
^
-^/3
-iVs
2
-fV3
315°
77r/4
-iV2
4V2
-1
-1
V^
-V2
330°
11W6
-I
iV^
-iv^
-v^
§V^
-2
345°
23^7/12
-i(\/6-V2)
i(-/6 + \/2)
-(2-\/3)
-(2 + VS )
■/6-\/2
-(\/6 + V2)
360°
2ir
1
q:(C
1
+ «=
For tables involving other angles see pages 206-211 and 212-215.
»■ — MT"
14
TRIGONOMETRIC FUNCTIONS
GRAPHS OF TRIGONOMETRIC FUNCTIONS
In each graph x is in radians.
5.22 y = sin a:
y
Fig. 5-5
5.24 y = tanx
Fig. 5-7
5.26
y = sec a;
J/
Fig. 5-9
5.23
y — cos X
Fig. 5-6
Fig. 5-8
5.27
y = CSC X
O
Fig. 5-10
FUNCTIONS OF NEGATIVE ANGLES
2:r
5.28 sin (—A) = —sin A
5.31 CSC (—A) = — c3Ci4
5.29 cos (— /4) = cosA
5.32 sec (—A) = sec A
5.30 tan(-A)
5.33 cot(-A)
— tan A
— cot A
TRIGONOMETRIC FUNCTIONS
16
ION formul:
5.34
5.35
5.36
5.37
sin (A^B) =
cos (A ± B) =
tan (A±B) =
cot (A^B) =
sin A cos B ± cos A sin B
cos A cos B m: sin A sin fi
tan A ± tan g
1 ^ tan A tan B
cot A cot B ^ I
cot A ± cotB
FUNCTIONS OF ANGLES IN AIL QUADRANTS IN TERMS OF THOSE IN QUADRANT I
-A
90^ ± A
- + A
2 -^
180^ i A
rr ± A
270"= It A
^±A
2
A:{360°) It A
2fcs- ± A
& — integer
sin
— sin A
cos A
51 sin A
— cos A
It sin A
cos
cos A
^ sin A
— cos A
± sin A
cos A
tan
— tan A
+ cot A
±tanA
ip cot A
-^ tan A
CSC
— CSC A
sec A
ip CSC A
— sec A
It CSC A
sec
sec A
~ CSC A
— sec A
± CSC A
sec A
cot
— cot A
wi tan A
ieotA
ip tan A
±cotA
RELATIONSHIPS AMONG FUNCTIONS OF ANGLES IN QUADRANT I
sin A — u
cos A — M
tan A = u
cot A = ti
sec A = u
CSC A ~ u
sin A
cos A
tan A
cot A
sec A
CSC A
u
1/m
w/Vl + "2
1/Vl + 1*2
Va2 - 1/m
1/u
Vi - ?(2
l/Vl + «2
1/u
j(/Vi + «2
1/m
u
\/u2 — l/z(
«/Vl - w^
Vi - «Vzt
y/v?-\
1/Vu2 - 1
V 1 ~ u^/u
m/Vi - «2
1/m
iNu"^ - 1
M
Vw^-l
1/Vl - m2
1/m
Vi + it^
Vl + W^M
m/Vm^ - 1
1/Vl - W2
Vl + uVu
Vl + M^
k/Vm2 - 1
Vl + «2
For extensions to other quadrants use appropriate signs as given in the preceding table.
16
TRIGONOMETRIC FUNCTIONS
5.38
5.39
5.40
sin2A
cos2A
tan2A
DOUBLE ANGLE FORMULAS
2 sin A cos A
cos^A — sin^A
2 tan A
1 - tan2 A
= 1 - 2 sin2 A = 2 cos^ A
r
5.41
5.42
5.43
. A
sin-
COSg
* A
tan-
HALF ANGLE FORMULAS
V
1 — cos A
1 + COS A
2
1 — cos A
1 + cos A
sin A
+ if A/2 is in quadrant I or II
— if A/2 is in quadrant III or IV
+ if A/2 is in quadrant I or IV
— if A/2 is in quadrant II or III
+ if A/2 is in quadrant I or III
— if A/2 is in quadrant II or IV
1 — cos A
1 + cos A
sin A
— esc A — cot A
MULTIPLE ANGLE FORMULAS
5.44
5.45
5.46
5.47
5.48
5.49
5.50
5.51
5.52
sinSA
cosSA
tanSA
sin4A
cos4A
tan 4A
sin&A
cosSA
tan5A
See also formulas 5.68 and 5.69.
3 sin A — 4 sin^ A
4 cos^ A — 3 cos A
3 tan A - tan^A
1-3 tan2A
4 sin A cos A — 8 sin^ A cos A
8 cos-* A - Scos^A + 1
4 tan A — 4 tan'* A
1-6 tan2 A + tan^ A
5 sin A - 20 sin^ A + 16 sin^ A
16 cos^ A — 20 cos^ A + 5 cos A
tanS A - 10 tan^ A + 5 tan A
1 - 10tan2 A + Stan^A
POWERS OF TRIGONOMETRIC FUNCTIONS
5.53 sin2A ^ ^-^cos2A
5.54 cos2A = | + ^cos2A
5.55 sin^A — £ sin A — ^ sinSA
5.56 cos'^fA — £cosA + icosSA
See also formulas 5.70 through 5.73.
5.57
sin^ A
5.58
cos^ A
5.59
sin^ A
5.60
cos^ A
^ — ^ COS 2A + ^ cos 4A
f + ^ cos 2A + ^ cos 4A
f sin A — ^ sin 3A + ^ sin 6A
; 3A + ^ cos 5A
§ cos-
^cos;
TRIGONOMETRIC FUNCTIONS
17
M, DIFFERENCE AND PRODUCT OF TRIGONOMETRIC FUNCTIO:4S
5.61 sin A + sin B - 2 sin ^(-4 + B) cos ^(A - B)
5.62 sin A - sin B = 2 cos ^(A + F) sin ^{A - B)
5.63 cos A + cos F = 2 cos ^(A + B) cos ^{A -^ B)
5.64 cos A - cos B = 2 3in^(A + B) sin i(B - A)
5.65 sin A sin B = ^{cos (A - B) - cos (A + B)}
5.66 cos A cos B = |{cos (A - B) + cos (A + B)}
5.67 sinAcosB = ^{sin (A - B) + sin (A + B)}
GENERAL FORMULAS
5.68 sinTiA ^ sinA ■j(2cosA)"-i - f " ^ ) (2 cos A)"-3 + f '^ 2 j (2 cos A)"-5 - ■ • ■ I
5.69 cosnA ^ |-^(2cosA)" - y{2cosA)"-2 + |-("~^j(2cosA)''-4
n In — 4
3 V 2
(2cosA)''-6 +
5.70 sin^n-iA =
5.71 cos2"-iA =
<g^|sin(2n-l)A - (^"^ ^ ] sin (2n - 3)A +
4)'
^-^^-^^t'-l'^^^"^
^A_|eos(2n-l)A + (^"^ ^)cos(2n-3)A + ••• + (^^_ ^^) cosaI
5.72 sin2nA =
5.73 cos2"A =
= 2^(1") +|2;r^|cos2nA -(2«)cos(2n-2)A + ■•- ("D-^ (^^^j )cos2A
h^l) + 2^f<'«2nA + (2n^ cos (2«-2)A +
+ ( ^ J* ^ ) cos 2A
If a: — sin J/ then y — sin~' x, i.e. the angle ■whose sine is x or inverse sine of x, is a many-valued
function of x which is a collection of single-valued functions called branches. Similarly the other inverse
trigonometric functions are multiple-valued.
For many purposes a particular branch is required. This is called the principal branch and the values
for this branch are called principal values.
18
TRIGONOMETRIC FUNCTIONS
NCiPAL VALUES FOR rNVERSE TRIGONOMETRIC FUNCTIO
Principal values for a; ^
Principal values for a- <
£ sin-' a; ^ W2
"W2 S sin-' a; <
^ cos-^iB ^ jr/2
5r/2 < cos-la: ^ tt
^ tan-la; < irll
— W2 < tan-' a; <
< cot-'a; g Tr/2
7r/2 < cot-ix < TT
^ see-' a: < jr/2
W2 < sec-' a; ^ -
< csc-'a: ^ jr/2
-7r/2 ^ csc-'x <
REt^TtON$ BETWEEN INVERSE TRIGONOMETRIC FUNCTIONS
In all cases it is assumed tbat principal values are used.
5.74 sin-' x + cos-' a: = 7r/2
5.75 tan-'x + cot-'ar ^ 7r/2
5.76 aec-'a: + esc-' a; = 57/2
5.77 CSC"' X — sin-' (1/a:)
5.78 sec-' 3; — cos-' (1/a;)
5.79 cot-' X = tan-' (1/a;)
5.80
sin-' (— fl!)
—
— sin-' X
5.81
COS-' {—x)
=
TT — COS-' X
5.82
tan-M-*)
=
— tan-' X
5.83
cot-' (-a:)
=
TT — cot-' X
5.84
sec-' (— x)
=
v — sec- ' *
5.85
CSC"' (—x)
=
— CSC-' X
INVERSE TRIGONOMETRIC
In each graph y is in radians. Solid portions of curves correspond to principal values.
5.86 y — sin-'x
5.87 y ~ cos-' a;
5.88 y = tan-'x
S-/2-
-W2 ■
Fig. 5-11
O :
V
y
^^
K
X
-I
A
/
/
/
1
; -'
■
y
V
7r/2
f X
_^
-H2
-""^
Fig. 5-12
Fig. 5-13
TRIGONOMETRIC FUNCTIONS
19
5.89 y = cot-^a;
Fig. 5-14
5.90 y = sec~'x
y
t/2
-1
-ir/2
O i\
5.91
Fig. 5-15
Fig. 5-16
RELATIONSHIPS BETWEEN SIDES AND ANGLES OF A PLANE TRIANGLE
The following results hold for any plane triangle ABC with
sides a, b, c and angles A, B, C.
5.92 Law of Sines
5.95
sin A sin B sin C
5.93 Law of Cosines
c2 = o2 + 62 _ 2a6 COS C
with similar relations involving the other sides and angles.
5.94 Law of Tangents , ,,^10*
a + h _ tan IJA + B)
a- b ~ tan ^(A - B)
with similar relations involving the other sides and angles.
sin A = T- y/a{8 — a)(8 — b){a — c)
Fig. 5-17
where s = |(a +b + c) is the semiperimeter of the triangle. Similar relations involving angles
B and C can be obtained.
See also formulas 4.5, page 5; 4.15 and 4.16, page 6.
RELATIONSHIPS BETWEEN SIDES AND ANGLES OF A SPHERICAL TRIANGLE
Spherical triangle ABC is on the surface of a sphere as shown
in Fig. 5-18. Sides a,b,c [which are arcs of great circles] are
measured by their angles subtended at center O of the sphere. A, B, C
are the angles opposite sides a,b,c respectively. Then the following
results hold.
5.96 Law of Sines
sin a sin b
sin A BinB sinC
5.97 Law of Cosines
cos o = cos b cos c + sin 6 sin c cos A
cos A = — cos B cos C + sin B sin C cos a
with similar results involving other sides and angles.
Fig. 5-18
]
20
TRIGONOMETRIC FUNCTIONS
5.98 Law of Tangents
tan ^(A + B) _ tan ^(a + b)
tan ^{A-B) ~ tan |(o - b)
with similar results involving other sides and angles.
5.99 cos— = ^I sin8sin(i
2 V sin 6 sii
where s = ^(a + 6 + c). Similar results hold for other sides and angles.
5.100
(s-c)
sin c
cos
a _ j cosjS-B) cos (S-C)
2 V sin 5 sin C
where S ~ ^(A + B + C). Similar results hold for other sides and angles.
See also formula 4.44, page 10.
NAPIER^-S^ RULES FOR RIGHT ANGIED SPHERICAL TRIANGLES
Except for right angle C, there are five parts of spherical triangle ABC which if arranged in the order
as given in Fig. 5-19 would be a,b,A,c,B.
co-B
co-A
co-c
Fig. 5-19
Fig. 5-20
Suppose these quantities are arranged in a circle as in Pig. 5-20 where we attach the prefix co
[indicating coviplement] to hypotenuse c and angles A and B.
Any one of the parts of this circle is called a middle part, the two neighboring parts are called
adjacent parts and the two remaining parts are called opposite parts. Then Napier's rules are
5.101 The sine of any middle part equals the product of the tangents of the adjacent parts.
5.102 The sine of any middle part equals the product of the cosines of the opposite parts.
Example: Since co-A = 90° -A, co-B = 90°— B. we have
sin a — tan b tan (co-B) or sin o — tan b cot B
sin (co-A) = cos a cos (co-B) or cos A = cos a sin B
These can of course be obtained also from the results 5.97 on page 19.
DEFINITIONS INVOLVING COMPLEX NUMBER
A complex number is generally written as a + bi where o and b are real numbers and i, called the
imaginary unit, has the property that v^ - ~1. The real numbers a and 6 are called the real and imaginary
parts oi a + bi respectively.
The complex numbers a + bi and a — bi are called complex conjugates of each other.
EQUALITY OF COMPLEX NUMBEI
6.1
a + bi — c + di if and only if a = c and b = d
6.2
{a+bii-i- ic + di) = {a+c) + ib + d)i
SUBTRACTION OF COMPLEX NUMBERS
6.3
(a + bi) - (c + di) = (a~c) + {b-d)i
MULTIPLICATION OF COMPLEX NUMBERS
6.4
(a+bi){c + di) = {ae - bd) + {ad + bc)i
DIVISION OF COMPLEX NUMBE
6.5
a + bi _ a + bi ^ c — di _ ac + bd f be — a,d\ .
c + di ~ c + di' c- di " c^ + d^ \ c'i + d? ' *
Note that the above operations are obtained by using the ordinary rules of algebra and replacing i^ by
—1 wherever it occurs.
21
22
COMPLEX NUMBERS
GRAPH OF A COMPLEX NUMBER
A complex number a + bi can be plotted as a point (a, b) on an
xy plane called an Argand diagram or Gaussian plane. For example
in Fig. 6-1 P represents the complex number —3 + 4i.
A complex number can also be interpreted as a vector from
O to P.
POLAR FORM OF A COMPLEX NUMBER
In Fig. 6-2 point P with coordinates (x, y) represents the complex
number x + iy. Point P can also be represented by pola/r coordinates
(r, e). Since x = r cos 6, y = r sin 6 we have
6.6
X -\- iy =^ r(cos 9 -V i sin e)
called the polar form of the complex number. We often call r = s/x^ + y^
the modulus and $ the amplitude of a; + iy.
Fig. 6-2
lUlTIPLICATION AND DIVISION OF COMPLEX NUMBERS IN POU^R FORM
6.7
6.8
[ri(cos tf 1 + 1 sin e{i\ [r2(cos $2 + * sin «2)] — »'i'"2[cos (*i + tfg) + ^ sin (ffj + e^i]
ri(cos ffi + 1 sin ffx) '"i
— ; : — : T — COS (81 — So) + * 31" (*1 ~ *2)l
r2(cos e-i + I sin 62) rg ^ ^ ^ ^ ^ ■"
DE MOIVRE'S THEOREMi|H
If p is any real number, De Moivre's theorem states that
6.9 [r(cos 9 A- i sin ff)]" = r''(cos pS + i sin p$)
ROOTS OF COMPLEX NUMBERS
6.10
If p = 1/n where n is any positive integer, 6.9 can be written
9 -I- 2fcff
[r(cos ff + t sin ff)] ^■''' = r^'
+ i sin
* + 2k
^]
where k is any integer. From this the n nth roots of a complex number can be obtained by putting
k = 0,1,2, . ..,n-l.
7
EXPONENTIAL and LOGARITHMIC
FUNCTIONS
7.2
a^laf^ = a""'
7.3
(aP)i = aP"
7.5
a-p = 1/aP
7.6
{q,6)p ^ apftp
7.8
"/ —
7.9
Va/6 = yfa/yfb
LAWS OF EXPONENTS
In the following p, q are real numbers and m, n are positive integers. In all cases division by zero is
excluded.
7.1 aP' ai = aP + 'i
7.4 qO = 1, o ?^
7.7 v^ - oi/«
In aP, p is called the exponent, a is the base and op is called the pth power of a. The function y = a^
is called an exponential function.
LOGARITHMS AND ANTILOGARITHMS
If aP = N where a # or 1, then p = logo N is called the logarithm ot N to the base a. The number
N = a'' is called the antilogarithm of p to the base a, written antilogy p.
Example: Since 3^ = 9 we have logs 9 — 2, antilog3 2 = 9.
The function y = log„x is called a logarithmic function.
LAWS OF LOGARITHMS
m^: i>m-i-l;^-mU-7''liiikV.9Ui v*p,1H^KS*«
7.10
7.11
7.12
log„ MA' ^ log„ M + log„ N
M
logo -^ - logo M - log„ A^
iogfl M^ = p logn M
COMMON LOGARITHMS AND ANTILOGARITHMS
Common logarithms and an ti logarithms [also called Briggsian] are those in which the base a — 10.
The common logarithm of N is denoted by logigN or briefly logN. For tables of common logarithms and
antilogarithms, see pages 202-205. For illustrations using these tables see pages 194-196.
23
24
EXPONENTIAL AND LOGARITHMIC FUNCTIONS
NATURAL LOGARITHMS AND ANTIIOGARITHMS
Natural logarithms and antilogarithms falso called Napierian] are those in which the base a= e =
2.71828 18 . . . [see page 1]. The natural logarithm of N is denoted by logg 'N or In A'. For tables of natural
logarithms see pages 224-225. For tables of natural antilogarithms [i.e. tables giving e^ for values of ic]
see pages 226-227. For illustrations using these tables see pages 196 and 200.
CHANGE OF BASE OF LOGARITHMS
The relationship between logarithms of a number 'N to different bases a and 6 is given by
log.iV
7.13
In particular,
7.14
7.15
logoiV =
logb a
loggiV = InN = 2.30258 50929 ...logioN
logjoN = logiV =: 0.43429 44819 ...log^iV
i
RELATIONSHIP BETWEEN EX
TIAL AND TRIGONOMETRIC FUNCT IONS
7.16 e^ = cos 9 + isins, e~'^ = cos ff — i sin ff
These are called Euler's identities. Here i is the imaginary unit [see page 21].
7.17
7.18
7.19 '
7.20
7.21
7.22
sin e
=
2i
cos tf
=
2
tan
=
cotff
=
-
2
gie + e-"
2i
e« -f e-'«
,w _ «-«
PERIODICITY OF EXPONENTIAL FUNCTIONS
7.23 e«»+2feTJ = fiW k = integer
From this it is seen that e* has period 2irx.
EXPONENTIAL AND LOGARITHMIC FUNCTIONS
25
POLAR FORM OF COMPLEX NUMBERS EXPRESSED AS AN EXPONENTIAL
The polar form of a complex number x + iy can be written in terms of exponentials [see 6.6, page 22]
*'*■* X + iy = r(cos a -\- i sin e) — re'S
OPERATIONS WITH COMPLEX NUMBERS IN POLAR FORM
Formulas 6.7 through 6.10 on page 22 are equivalent to the following.
7.26
7.27
7.28
r2e'"2 r2
(rei«)p = r^e'pfl [De Moivre's theorem]
LOGARITHM OF A COMPLEX NUMBER
7.29
In (re») = \nr + ie + 2kwi k = integer
DEFINITION OF HYPERBOLrc FUNCtlONS
8.1
8.2
8.3
8.4
8.5
8.6
Hyperbolic sine of a; = sinh x =
Hyperbolic cosine of a; = cosh x =
Hyperbolic tangent of z = tanh z —
Hyperbolic cotangent of x = coth x —
Hyperbolic secant of a; = sech x =
Hyperbolic cosecant of x = csch x =
gX
— e~
X
2
^ + e-
-X
2
e^
— e
-X
e^ + e-
-X
e^
+ e'
-E
e^
— e'
2
-I
e^
+ e-
2
-X
dI _ P--I
RELATIONSHIPS AMONG HYPERBOLIC FUNCTIONS
8.7
8.8
8.9
8.10
8.11
8.12
8.13
sinh a;
coshic
1
=
coshsr
tanh X
sinhx
sechx
1
cosh X
csch a;
1
sinh a;
cosh^ X
— sinh^ X —
1
sech^ a;
+ tanh2 X =
1
C0th2 X
— csch^ X =
1
FUNCTIONS OF NEGATIVE ARGUMENTS
8.14 sinh (—a:) — — sinh a:
8.17 csch (— ic) = —csch a:
8.15 cosh(— «) = cosh a:
8.18 aech(-a;) = sech a;
8.16 tanh (—a:) = — tanha;
8.19 coth (-a;) - -cotha;
26
HYPERBOLIC FUNCTIONS
27
8.20
8.21
8.22
8.23
sinh {x ± y)
cosh {x ± y)
tanh (x ± y)
DITION FORMUiAS
sinh X cosh y ± cosh x sinh y
cosh X cosh v ± sinh x sinh i/
tanh X ± tanh |/
1 ± tanh a; tanh j/
coth ix±y) =
coth z coth y ± 1
coth v ± coth X
DOUBLE ANGLE FORMULAS
8.24
sinh 2x
= 2 sinh X cosh x
8.25
cosh 2x
= cosh^x + sinh^x
8.26
tanh 2a;
2 tanhx
1 + tanh'* X
= 2 cosh2 X - 1 = 1 + 2 sinh2 x
8.27
8.28
8.29
HALF ANGLE FORMULAS
sinh I
cosh:
tanh I
cosh X — 1
cosh X + 1
cosh X — 1
[+ if X > 0, - if X < 0]
\ cosh X + 1
sinh X cosh x — 1
[+ if X > 0, - if X < 0]
cosh X + 1
sinhx
8.30
8.31
8.32
8.33
8.34
8.35
sinh 3x =
cosh 3x —
tanh 3x =
sinh 4x =
cosh 4x =
tanh 4x —
3 sinh X + 4 sinh^ x
4 cosh^ X — 3 cosh x
3 tanh x + tanh^ x
1 + 3 tanh2 x
8 sinh^ X cosh x + 4 sinh x cosh x
8 cosh'' X — 8 eosh2 x + 1
4 tanh x + 4 tanh-^ x
1 + 6 tanh2 x + tanh* x
28
HYPERBOLIC FUNCTIONS
8.36
8.37
8.38
8.39
8.40
8.41
sinh" X
—
^ cosh 2ar — ^
cosh* X
-
1 cosh 2x + ^
sinh^ a:
-
i sinh 3x — 1 sinh a
cosh^ X
-
^ cosh 3a; + a cosh a;
sinh^ X
=
1^ — ^ cosh 2x + ^ cosh 4a;
cosh* X
=
f + ^ cosh 2x + ^ cosh 4x
8.42
8.43
8.44
8.45
8.46
8.47
8.48
SUM, DIFFERENCE AND PRODUCT OF HYPEl
sinh X + sinh i/ = 2 sinh ^{x + y) cosh ^(x — y)
sinh X - sinh y = 2 cosh ^(x + y) sinh ^(x - y)
coshx + coshy = 2 cosh ^(x + y) cosh ^(x — j/)
cosh X ~ cosh y = 2 sinh ^{x + y) sinh ^(x — y)
sinh X sinh y = ■J{cosh ix + y) - cosh {x - y))
cosh X cosh y — ^{cosh (x + y) + cosh (x — y))
sinh X cosh y — ^{sinh {x + y) + sinh (x — y)}
XPRESSION OF HYPERBCtIC FUNCTIONS IN TERMS OF OTHERS
In the following we assume x > 0. If x < use the appropriate sign as indicated by formulas 8.14
to 8.19.
sinh X = a
cosh X = M
tanh X — w
coth X = u
sech X ~ M
csch X — M
sinh X
coshx
tanh X
coth X
sech X
csch X
u
1/m
V«2-l
u
m/V1 - u^
1/Vk2 - 1
Vl - "2/m
l/7(
Vl + W2
1/Vl - m'^
M
1/m
m/Vm2 - 1
1/m
M
Vl + uVu
k/Vi + «2
^/u^ - llu
Vl-w-
1/Vl + m2
V"'' + 1/m
uHv?- - 1
1/m
l/\/l - m2
u
\/l + "^
1/m
Vi-«2
Vm2 - 1/m
m/V1 + m2
M
I/Vk^ - I
Vi - uV«
Vw2 - 1
«/Vl - m^
HYPERBOLIC FUNCTIONS
29
)F HYPERBO
8.49
Fig. 8-1
8.50
y = cosh X
O
Fig. 8-2
8.5 i
y = tanh x
Fig. 8-3
8.52 y = coth X
y
Fig. 8-4
8.54 y = csch X
y\
Fig. 8-5
Pig. 8-6
JNVERSE HYPERBOLIC FUNCTIONS
I^ X = sinhj/, then y = sinh~'x is called the inverse hyperbolic sine of x. Similarly we define the
other inverse hyperbolic functions. The inverse hyperbolic functions are multiple-valued and as in the
case of inverse trigonometric functions [see page 17] we restrict ourselves to principal values for which
they can be considered as single-valued.
The following list shows the principal values [unless otherwise indicated] of the inverse hyperbolic
functions expressed in terms of logarithmic functions which are taken as real valued.
8.55
8.56
8.57
8.58
8.59
8.60
sinh->a; = In (x + y/x^ + 1 )
cosh""! X — In (a; + ^/x^ — 1 )
tanh ^ X = — In ( :;
2 \1 — X
coth * X = - in — --
2 \x — 1
c«:h->x = In(^^+ -^^+1
— » < « < =0
x^l [cosh^^x > is principal value]
-1 < X < 1
X > 1 or X < — 1
< X S 1
x^
[sech~' X > is principal value]
30
HYPERBOLIC FUNCTIONS
8.61
8.62
8.63
8.64
8.65
8.66
8.67
RELATIONS BETWEEN INVERSE HYPE^G^fcFUNCTIONS
esch-'a; = sinh~^(l/a:)
sech-^a: = cosh^i(l/a:)
coth-ix = tanh"' (1/a;)
sinh~i {—x) = — sinh-i x
tanh~*(— a;) = — tanh^'a;
coth^i {—x) = — coth~' X
csch^* (— ar) — — cseh~' x
RAPHS OF INVERSE HYPERBOLIC FUNCTIONS
8.69 y = cosh -la:
Fig. 8-8
8.72 y - sech-ia:
V
O
II
Fig. 8-10
Fig. 8-11
Fig. 8- J 2
HYPERBOLIC FUNCTIONS
31
RELATIONSHIP BETWEEN HYPERBOLIC AND TRIGONOMETRIC FUNCTIONS
8.74 sin (ix) = i sinh z
8.77 CSC {ix) = — i csch x
8.80 sinh [ix) — i sin x
8,83 csch (ix) = — i CSC x
8.75 cos [ix) — cosh x
8.78 sec (ix) = sech x
8.81 cosh (ia;) — cos a;
8.84 sech (ix) — sec x
8.76 tan (ix) — i tanh x
8.79 cotii'x) — — icothx
8.82 tanh {ix) = i tan x
8.85 coth (ia:) = —i cot a:
PERIODICITY OF HYPERBOLIC FUNCTIONS
In the following k is any integer.
8.86 smh(x4-2ijrt") = sinhx 8.87 cosh(x + Zkin) ^ cosh a: 8.88 tanh{x + kvi) = tanh a:
8.89 csch (a; + 2kiri) ^ csch x 8.90 sech {x + 2kTri) = sech x 8.91 coth (a; + kiri) = coth a;
RELATIONSHIP BETWEEN INVERSE HYPERBOLIC AND INVERSE TRIGONOMETRIC FUNCTIONS
8.92 sin~i(ix) = i sinh"' a;
8.94 cos ~^ X =■ ± i cosh ~ ' x
8.96 tan-i(iar) = i tanh -'a
8.98 cot"''(ta;) — — ieoth~'a:
8.100 sec~'x = ±isech~^a;
8.102 csc~' {ix) ^ ~i csch~^ x
8,93 sinh~i {ix) = i sin~' x
8.95 cosh"' a: = ±icos~^x
8.97 tanh-'(ix) = itan-'a:
8.99 coth-'(ia:) = -icot-ix
8.101 sech~^ X = ±i sec~^ x
8.103 cseh~'(ta;) — — icsc~ix
QUADRATIC EQUATION: ax- -h bx + c =
9.1
Solutions :
-6 ± \/b^ - iac
2a
9.2
If a, b, c are real and if D = b^ — 4a« is the discriminant, then the roots are
{i) real and unequal if D >
(ii) real and equal if D =
(iii) complex conjugate if Z? <
If x^, X2 are the roots, then Xi + arg — —b/a and a;,a-2 — c/a.
Ee
CUBIC EQUATION; x'^ + a,x^ -\- a-iZ + a^ =
L^
Q =
Za^ ~ (i\
R =
9aja2 — 27a3 — 2ai
9 ' " 54
a:, = S + r - lai
9.3 Solutions: -j ajg = -^(S + T) - ^Oi + ^i-/3 (S - T)
3^3 = -i(S + r) - !^a, - \iVZ(S - T)
If ttj, 03,03 are real and if D = Q'^ + R'^ is the discriminant, then
(i) one root is real and two complex conjugate if D >
(ii) all roots are real and at least two are equal if Z> =
(iii) all roots are real and unequal if D < 0.
If D < 0, computation is simplified by use of trigonometry.
9.4
93
Solutions if Z> < 0:
Xi - 2\/— Q cos (J*)
Xz = 2-/^ cos (^« + 120°)
23 = 2^/^cos(^tf + 240°)
where cos S = —R/yJ—Q^
where Xi,xi,,x^ are the three roots.
= -a-i
32
SOLUTIONS OF ALGEBRAIC EQUATIONS
33
* B^ BM I I ■ iiii^ii l i I i llllllM^^BTCT i l ll ll il llll fcii
QUARTIC EQUATION: x* + a,*^ + azX^ + CsX + a4 =
Let yi be a real root of the cubic equation
9.6 y3 ~ 02^2 -I- (a^ag - 404)1/ + (40304 -al- ofo^) -
9.7 Solutions: The 4 roots of z^ + ^{a, ± y/a\ - ia^ + Ay^ )z + ^{j/j ± y/y\ - 404 } =
If all roots of 9.6 are real, computation is simplified by using that particular real root which produces
all real coefficients in the quadratic equation 9.7.
9.8
Kj + a^a + ajg + 3:4 = — Oi
XiX2 + X2ir3 + X^X^ + X^Xi + X^X^ + X2X^ = a2
XiX2X^ + ^2X3X4 + 9;j!E2-'^4 "I" ^l^Z-'^i ^^ — *^3
3:1X2X3X4 = 04
where Xj, Xg, Xg, X4 are the four roots.
10
FORMULAS from
PLANE ANALYTIC GEOMETRY
DISTANCE d BETWEEN TWO POINTS Piixx,yi) AND ^2(2^2,^2)
10.1
d = V (3^2 -Xi)2 + (j/2- 1/1)2
:iL
P,(a;i,!/i)
l(3/2-!/i)
1^2- X,)
-1
Xi
Fiff. 10-1
HOPE m OF LINE JOINING TWO POINTS Piixi,yi) AND P2{x.z,yz)
10.2
V2 - Vi
m = = tariff
X2 - «!
UATION OF LINE JOINING TWO POINTS Pi{Xi,yi) AND Pz{x2,y2)
10.3
10.4
y-y\ _ vz- vi
X — Xi x^— Xi
m or y — Vi = «*(« — a^i)
y = mx + 6
XzVi — Xiifz
where =^ y.— mx, = is the intercept on the y axis, i.e. the y intercept.
X2 - i'l
lEQUATION OF LINE IN TERMS OF x INTERCEPT a^O AND y INTERCEPT b ¥=
10.5
- + ^ - 1
a b
Fig. 10-2
34
FORMULAS FROM PLANE ANALYTIC GEOMETRY
35
10.6 xcosa + ysina — p
where p = perpendicular distance from origin O to line
and a = angle of inclination of perpendicular with
positive X axis.
Fig. 10-3
GENERAL EQ
10.7
Ax + By + C =
DISTANCE FROM POINT {xi,yi) TO LINE Ax + By + C =
10.8
Axj + Byi + C
where the sign is chosen so that the distance is nonnegative.
LNGLE ^ BETWEEN TWO LINES HAVING SLOPES mi AND Mz
10.9
tan^ =
Wl2 ~ Wi
1 + mim2
Lines are parallel or coincident if and only if wii = m2.
Lines are perpendicular if and only if m2 =■ — iMi-
Fig. 10-4
slope Bii
slope nij
AREA OF TRIANGLE WITH VERTICES AT {a;i, ?/i), (:r2, 3/2), (a^a. ^s)
10.10 Area = ±
Xi Vi 1
X2 i/2 1
X3 Va 1
ix2, j/a)
where the sign is chosen so that the area is nonnegative.
If the area is zero the points all lie on a line.
(*i.Vi)
Fig. 10-5
36
FORMULAS FROM PLANE ANALYTIC GEOMETRY
rRANSFORMATION OF COORDINATES INVOLVING PURE TRANSLA
10.11
X = x' + X(i
y - y' + vq
JX' = X ~ Xq
{y' = y - Vq
where {x, y) are old coordinates [i.e. coordinates relative to
xy system], {x',y') are new coordinates [relative to x'y' sys-
tem] and (aro.tfo) are the coordinates of the new origin O'
relative to the old zy coordinate system.
O
y'
ixa-Vii)
O'
Fig. 10-6
TRANSFORMATION OF COORDINATES INVOLVING PURE ROTATION
x'
10.12
ic — a; cos a — 1/ sin a (x' — x cosa + y nma
_ , , or -^
y — X sma + y'cosa yy' = ?/ cos a — x sin a
where the origins of the old [xy] and new [x'y'] coordinate
systems are the same but the x' axis makes an angle a with
the positive x axis.
.y'
0^
\
Fig. 10-7
TRANSFORMATION OF COORDINATES INVOLVING TRANSLATION AND ROTATION
10.13
X = x' cos a — y' sin a + xq
V = x' sin a + y' cos a + y^
x' = (x — Xq) cos a + iy — yo) sin a
y' = [y — Vo) cos a — (x — x^) sin a
where the new origin O' of x'y' coordinate system has co-
ordinates (xQ,yQ) relative to the old xy coordinate system
and the x' axis makes an angle a with the positive x axis.
-^
\
\
o \
\
\
Fig. 10-8
x'
A point P can be located by rectangular coordinates (x,^) or
polar coordinates (r, »). The transformation between these coordinates
Is
10.14
X = r cos *
y — r sin $
r = V*^ + y^
= tan-^ (y/x)
O
Fig. 10-9
FORMULAS FROM PLANE ANALYTIC GEOMETRY
37
EQUATION OF CIRCLE OF RADIUS R, CENTER AT [x^yo)
10.15
(a!-Xo)2+(wo)2 = fl2
Fig. 10-10
IQUATION OF CIRCLE OF RADIUS R PASSING THROUGH ORI6I
10.16 r = 2Rcos(s-a)
where (r, e) are polar coordinates of any point on the
circle and (R,a) are polar coordinates of the center of
the circle.
Fig. 10-11
CONICS [ELLIPSE, PARABOLA OR HYPERBOLA]
If a point P moves so that its distance from a fixed point
[called the focus] divided by its distance from a fixed line [called
the directrix] is a constant e [called the eccentricity], then the
curve described by P is called a conic [so-called because such
curves can be obtained by intersecting a plane and a cone at
diflFerent angles].
If the focus is chosen at origin O the equation of a conic
in polar coordinates (r, e) is, if OQ = p and LM = D, [see
Fig. 10-12]
10.17
P
tD
1 — e cos e
The conic is
(i) an ellipse if « < 1
(ii) a parabola if « = 1
(iii) a hyperbola if e > 1.
1 — e cos 8
Directrix
Fig. 10-12
38
FORMULAS FROM PLANE ANALYTIC GEOMETRY
ELLIPSE WITH CENTER C(Xo, 1/0) AND MAJOR AXIS PARALLEL TO x
10.18 Length of major axis A'A — 2a
10.19 Length of minor axis F'B = 26
10.20 Distance from center C to focus F or F' is
= Va2 - 62
10.21 Eccentricity ^
v^airp
O
10.22 Equation in rectangular coordinates:
{a: - Xo)g {y - Vp)^
n "r 10 — 1
62
10.23
Fig. 10-13
Equation in polar coordinates if C is at O: r^ = -r —
a2 sin2 9 + 0* cos2 6
10.24 Equation in polar coordinates if C is on « axis and F' is at O: r — -^ — ^
1 — « cos e
10.25 If P is any point on the ellipse, PF + PF' - 2a
If the major axis is parallel to the y axis, interchange x and j/ in the above or replace e by iir
90° - e].
e [or
PARABOLA WrTH AXIS PARALLEL TO X AXIS
If vertex is at A{xQ,yff) and the distance from A to focus F is a > 0, the equation of the parabola is
10.26 iy — yo)^ = ^a{z — XQ) if parabola opens to right [Fig. 10-14]
10.27 {y~yo)^ = — 4a{ic-a;o) if parabola opens to left [Fig. 10-15]
If focus is at the origin [Fig. 10-16] the equation in polar coordinates is
10.28
2a
1 — cos 9
Fig. 10-14 Fig. 10-15 Fig. 10-16
In case the axis is parallel to the y axis, interchange x and y or replace 8 hy ^ir — 8 [or 90° — e].
FORMULAS FROM PLANE ANALYTIC GEOMETRY
PERBOIA WITH CENTERX^o^yo) AND^AJ OR AJOSPll^rLilTOT^
«'n.
^H
Fig. 10-17
10.29 Length of major axis A'A = 2a
10.30 Length of minor axis B'B = 26
10.31 Distance from center C to focus F or F' = e = yaf+b^
C V^2T62
10.32 Eccentricity e = — =
a a
10.33 Equation in rectangular coordinates:
(g - iCoP {y - yg)^
a2 62
= 1
10.34 Slopes of asymptotes G'H and Gff' = ±~
10.35 Equation in polar coordinates if C is at O: r^ =
a262
62 cos^ tf — a2 sin2 g
10.36 Equation in polar coordinates if C is on X^ axis and F' is at O: r = - °^ ~ — ii-
1 — E cos S
10.37 If P is any point on the hyperbola, PF ~ PF' = ±2a [depending on branch]
If the major axis is parallel to the y axis, interchange x and y in the above or replace e by i^r — e
for 90° - el
LSMNfSCATE
11.1 Kquation in polar coordinates:
7-2 — Or cos 2tf
11.2 Equation in rectangular coordinates:
11. 3 Angle between AB' or A'B and x axis ~ 45''
1 1 .4 Area of one loop — a^
V
As
/^
~
^
(^%
/
^
\
A'y
B'
Fig. I 1-1
CYCLOID
1 1 .5 Equations in parametric form:
X — a(0 — sin 0)
— a{l — COS0)
11.6 Area of one arch = Swa^
1 1 .7 Arc length of one arch = 8a
This is a curve described by a point P on a circle of radius
a rolling along x axis.
Fig. 11-2
HYPOCYCLOID WITH FOUR CUSPS
11.8 Equation in rectangular coordinates:
n .9 Equations in parametric form:
X — a cos** 8
y = a sin'^ 8
11.10 Area bounded by curve = ■f"''*^
11.11 Arc length of entire curve — 6a
This is a curve described by a point P on a circle of radiiis
a/4 as it rolls on the inside of a circle of radius a.
Fig. 11-3
40
SPECIAL PLANE CURVES
41
11.12 Equation: r — a(l + cos tf)
1 1.13 Area bounded by curve = j^a^
1 1.14 Arc length of curve = 8a
This is the curve described by a point P of & circle of radius
a as it rolls on the outside of a fixed circle of radius a. The
curve is also a special case of the limacon of Pascal [see 11.32].
lATENARY
11.15 Equation: y = -{e^^'^+e^'^") = acosh-
2 o,
This is the curve in which a heavy uniform chain would
hang if suspended vertically from fixed points A and B.
THREE-LEAVED ROSE
11.16 Equation: r = a cos 3«
The equation r = a sin 3s is a similar curve obtained by
rotating the curve of Fig. 11-6 counterclockwise through 30^^ or
Tr/6 radians.
In general r — a cos ns or r = a sin n8 has n leaves if
n is odd.
Fig. n-4
Fig. ll-«
11.17 Equation: r = a cos 2tf
The equation r = a sin 2e is a similar curve obtained by
rotating the curve of Fig. 11-7 counterclockwise through 45° or
w/4 radians.
In general r — a cos ntf or r ~ a sin ne has 2n leaves if
n is even.
Fig. 11-7
42
SPECIAL PLANE CURVES
11.18 Parametric equations:
K = (a + 6) COS 8 — b cos ( — -. — )
y = (a + 6) sin tf — 6 sin ( — ; — I e
This is the curve described by a point P on a circle of
radius 6 as it rolls on the outside of a circle of radius a.
The cardioid [Fig. 11-4] is a special case of an epicycloid.
Fig. 11-8
11.19 Parametric equations:
X = (a—b) cos + 6 cos [ — z — ) *
y = {a—b) sin ^ — 6 sin [ — j — ] ^
This is the curve described by a point P on a circle of
radius b as it rolls on the inside of a circle of radius a.
If 6 = a/4, the curve is that of Fig. 11-3.
Fig. 11-9
TROCHOID
1 1 .20 Parametric equations:
X = atp — b sin <p
y = a — b cos
This is the curve described by a point P at distance 6 from the center of a circle of radius a as the
circle rolls on the x axis.
If b < a, the curve is as shown in Fig. 11-10 and is called a curtate cycloid.
If b > a, the curve is as shown in Fig. ll-H and is called a prolate cycloid.
If & = a, the curve is the cycloid of Pig. 11-2.
Fig. 11-10
Fig.U-U
SPECIAL PLANE CURVES
43
11.21 Parametric equations:
a; — a In (cot ^^ — cos ^)
y — a sin
This is the curve described by endpoint P of a taut string
PQ of length a as the other end Q is moved along the x
axis.
WITCH OF AGNESI
8aS
11.22 Equation in rectangular coordinates: y = ■ „ -l 4„2
1 1 .23 Parametric equations:
X — 2a cot e
y — a{\ — cos 2fl)
In Fig. 11-13 the variable line OA intersects y = 2a
and the circle of radius a with center (0, a) at A and B
respectively. Any point P on the "witch" is located by con-
structing lines parallel to the x and y axes through B and
A respectively and determining the point F of intersection.
Fig. 11-13
FOLIUM OF DESCARTES
11.24 Equation in rectangular coordinates;
3[;3 4- ^3 = Zaxy
1 1 .25 Parametric equations:
V =
Sat
1 + (3
3at^
1 + (3
11.26 Area of loop
= ^nZ
1 1.27 Equation of asymptote: x + y + a =
Fig. 11-14
INVOLUTE OF A CIRCLE
1 1 .28 Parametric equations:
x = a(cos + sin 0)
y = a(sin — cos 0)
This is the curve described by the endpoint P of a string
as it unwinds from a circle of radius a while held taut.
Fig. 11-15
44
SPECIAL PLANE CURVES
Equation in rectangular coordinates:
(oa;)2/3 + (6y)2/3 = (a2 _ 62)2/3
11.29
1 1 .30 Parametric equations:
ax = (a^ ~ 62) cogS g
by = {a2 - 62) ginS g
This curve is the envelope of the normals to the ellipse
xya^ + y^/b^ = 1 shown dashed in Fig. 11-16.
Fig. 11-16
11.31 Polar equation: r* + a* - 2aV2 cos 29 = 6*
This is the curve described by a point P such that the product of its distances from two fixed points
[distance 2a apart] is a constant b^.
The curve is as in Fig. 11-17 or Fig. 11-18 according as 6 < a or b > a respectively.
If 6 = o, the curve is a lemniaeate [Fig. 11-1].
^^i
^
Fig. 11-17
Fig. 11-18
:ON OF PASCAl
1 1 .32 Polar equation; r = b ■¥ a cos 8
Let OQ be a line joining origin O to any point Q on a circle of diameter a passing through O. Then
the curve is the locus of all points P such that PQ = 6.
The curve is as in Fig. 11-19 or Fig. 11-20 according as 6 > a or 6 < a respectively. If b = a, the
curve is a cardioid [Fig. 11-4].
y
Fig. 11-19
Fig. n-20
SPECIAL PLANE CURVES
45
CISSOID OF DIOCIES
11.33 Equation in rectangular coordinates:
y2 =
2a — X
11.34 Parametric equations:
fx — 2a sin^ 9
V -
2a sin^ 9
cos 9
This is the curve described by a point P such that the
distance OP = distance RS. It is used in the problem of
duplication of a cube, i.e. finding the side of a cube which has
twice the volume of a given cube.
Fig. 11-21
^SPIRAL OF ARCHIMEDES
11.35 Polar equation: r — a9
Fig. 11-22
12
FORMULAS from SOLID
ANALYTIC GEOMETRY
DISTANCE d BETWEEN TWO POINTS Pi{xi,yuZi) AND P%{x2,%2,Z2)
12.1 d = \f(x^ - a;,)2 + (ya - y,)2 + (z^ - ^^)2
Pz(^2.y2'^2)
FifT. 12-1
DIRECTION COSINES OF LINE JOINING POINTS P^{xx,yi,z,) AND >2(i2, 1/2,^2)
12.2
i — COS a = -t , m = cos/3 = , n — cos y —
add
where a,/?,y are the angles which line PjPa makes with the positive x,y,z axes respectively and
d is given by 12.1 [see Fig. 12-1].
RELATIONSHIP BETWEEN DIRECTION COSINES
12.3
cos2 ct + cos^ j8 + cos2 7 = 1 or I- + nfi -\- v?- = \
DIRECTION NUMBERS
Numbers L,M,N which are proportional to the direction cosines l,m,n are called direction numbers.
The relationship between them is given by
12.4
I ^
M
N
y/L^ + M^ + m \/lJ+~M^Tm' ^/UVmTm
46
FORMULAS FROM SOLID ANALYTIC GEOMETRY
47
EQUATIONS OF LINE JOINING Pi{xuyuZi) AND ^2(^2, ?/2, 22) IN STANDARD TORM
12.5
X - xi y - j/i z - z, X- Xi y - y\ z — z^
or
These are also valid if l,m,n are replaced by L,M,N respectively.
EQUATIONS OF LINE JOINING Pi{xi,yuZi) AND ^2(^2, ?/3. 22) IN PARAMETRIC FORM
12.6 X = X] -^ It, y — ]/] + mi, z — zj + wt
These are also valid if t,m,n are replaced by L,M,N respectively.
ANGLE BETWEEN TWO LINES WITH DIRECTION COSINES U.muni AND ;2,m2,«2
12.7
cos — ^1^2 + Wimg + »liW.2
GENERAL EQUATION OF A PLANE
12.8
Ax + By + Cz + D =
[A,B,C,D are constants]
EQUATION OF PLANE PASSING THROUGH POINTS {Xi,yi,Zi), (3^2,3/2,22), (3:3,1/3,23)
12.9
12.10
X — xi y - ifi z - zi
=
or
^2 - yi ^2 - 2l
(x-x^) +
Zg ^ Zi Xa — Xi
H ~ ^1 ^3 ^ a^i
iy-yi) +
X2 - xi vz- y-i
X3 - Xi yg - yi
(z - zi) =
EQUATION OF PLANE IN INTERCEPT i^ORM
12.11
a 6 c
where a,b,c are the intercepts on the x,y,z axes
respectively.
Fig. 12-2
48
FORMULAS FROM SOLID ANALYTIC GEOMETRY
12.12
EQUATIONS OF LINE THROUGH (xo, ?/o, So)
AND PERPENDICULAR TO PLANE Ax + By -\- Cz + D =
^ — a^o y - yo z~ ^0
or X — Xq + At, y = Va + Bt, z = Zq + Ct
Note that the direction numbers for a line perpendicular to the plane Ax + By + Cz + D = are
A,B,C.
DISTANCE FROM POINT {xo,yo,z,y) TO PLANE Ax + By -^ Cz -\- D =
12.13
Axo + Byo + Czo + D
where the sign is chosen so that the distance is nonnegative.
12.14
NORMAL FORM FOR EQUATION OF PLANE
X COS a + y cos j3 + 2 cos y = p
where p = perpendicular distance from to plane at
P and a,p,y are angles between OP and positive x,y,z
axes.
Fig. 12-3
TRANSFORMATION OF COORDINATES INVOLVING PURE TRANSLATION
12.15
X = x' + Xq
y - y' + Vo
Z - Z' -\- Zq
or
= z — Za
where (x, y, z) are old coordinates [i.e. coordinates rela-
tive to xyz system! , {x',y',z') are new coordinates [rela-
tive to x'y'z' system] and (kqi l/o-'^o) ^^^ ^^^ coordinates
of the new origin O' relative to the old xyz coordinate
system.
A
\z'
Jjfo^o^^o)^ _ ,
Fig. 12-4
FORMULAS FROM SOLID ANALYTIC GEOMETRY
49
12.16
TRANSFORMATION OF COORDINATES rNVOLVING PURE ROTATION
X = l^x' + I2V' + I3Z'
y = Ttiix' + m^y' + m^z'
z = n,r' + n2V' + TI32'
x' = lix + m-iy + riiz
y' - I2X + m^y + 7132
s' - l^x + m^y + Mgz
where the origins of the xyz and x'y'z' systems are the
same and /i,7ni,?i,; i2.»«2>"2'> '3.'«3."3 are the direction
cosines of the x',y',z' axes relative to the x,y,z axes
respectively.
Fig. 12-5
TRANSFORMATION OF COORDINATES INVOLVING TRANSLATION AND ROTATION
12.17
= hx' + l^y' + hz' + aro
— «,^' -J- ™_„' -f ^^2'
+ ngz' + Zo
a; = l^x' + /gy' + l^z' +
1/ = Wia;' + m2y' + wigz
+ I/O
y' = ya: - Xq) + m2{y - yg) + n^iz - «o)
^ 2' = ^3(3; - »(,) + ?n3(y - yo) + 713(2 - Zq)
where the origin O' of the x'y'z' system has coordinates
(a^o. Vd' ^o) relative to the xyz system and ly, m^, jj,;
^2>*"2."2! h''^S'i^z ^^^ the direction cosines of the
x',y',z' axes relative to the x,y,z axes respectively.
/
/
T
CYIINDRICAL COORDINATES {r,e,z)
Fig. 12-6
A point P can be located by cylindrical coordinates (r, 9, z)
[see Fig. 12-7] as well as rectangular coordinates (ir.y, 2).
The transformation between these coordinates is
12.18
X = r cos
y = r sin (
z = z
r — y/x^ + j/2
e — tan- 1 {yfx)
z = z
(^.tf.z)
Fig. 12-7
m
FORMULAS FROM SOLID ANALYTIC GEOMETRY
SPHERICAL COORDINATES {r,e,,p)
A point P can be located by spherical coordinates (r,e,
[see Fig. 12-8] as well as rectangular coordinates (x, y, z).
The transformation between those coordinates is
12.19
<ar
X = r sm e cos <f>
y = r sin e sin
z — r cos e
'r = V^2T^2 + ^
■^ = tan^i (j//ar)
EQUATION OF SPHERE IN RECTANGULAR COORDINATES
1 2.20 (X - xo)2 +{v- vo)^ + (z- zo)2 = R^
where the sphere has center {xq, i/q, zq) and radius R.
Fig. 12-9
EQUATION OF SPHERE IN CYLINDRICAL COORDINATES
12.21 7-2 _ 2rorcos{o-9Q) + r^ + (z-Zq)^ = R^
where the sphere has center (r(,,tfo,«o) in cylindrical coordinates and radius R.
If the center is at the origin the equation is
12.22 r^ + z^ = iJ2
EQUATION OF SPHERE IN SPHERICAL COORDINATES
12.23 r^ + To — Zr^r sin 9 sin So cos (^ - <Pq) = R~
where the sphere has center {7*o, 6q, <Pq) in spherical coordinates and radius R.
If the center is at the origin the equation is
12.24 r = R
FORMULAS FROM SOLID ANALYTIC GEOMETRY
51
EQUATION OF ELLIPSOID WITH CENTER (xo,yo,2o) AND SEMI-AXES a,b,C
a^ b^ c^
Fig. 12-10
ELLIPTIC CYLINDER WITH AXIS AS z AXIS
12.26
where a, 6 are semi-axes of elliptic cross section.
If h — a it becomes a circular cylinder of radius a.
12.27
Fig. 12-n
riC CONE WITH AXIS AS z AXIS
a2 ^ 62
HYPERBOLOID OF ONE SHEET
Fig. 12-12
12.28
^ A. y^ - ^ - 1
a2 "^ 62 c2
Fig. 12-13
52
FORMULAS FKOM SOLID ANALYTIC GEOMETRY
PERBOLOID OF TWO SHEET
12.29
a2 62 c2 ~
Note orientation of axes in Fig, 12-14.
Fig. 12-14
ELLIPTIC PARABOLOI
12.30
x2 ^
a2 "^ 62
Fig. 12-15
12.31 4-tf = -
Note orientation of axes in Pig. 12-16.
Fig. 12-16
1
DEFINITION OF A DERIVATIVE
If 1/ - f{x), the derivative of y or f(x) with respect to 2 is defined as
13 1 ^ = lim /<" + fc)-^<'> = lim /(^ + ^^)-/(^)
da; ji-o ft Ax-*o Az
where h = ^x. The derivative is also denoted by y' , df/dx or f'{x). The process of taking a derivative is
called differentiation.
GENERAL RULES OF DIFFERENTIATION
In the following, u, v, w are functions of x; a, b, c, n are constants [restricted if indicated]; e = 2.71828 . . .
is the natural base of logarithms; In m is the natural logarithm of u [i.e. the logarithm to the base e] where
it is assumed that a > and all angles are in radians.
13.2 £(c) =
13.3 4-i<^=^) = «
ax
13.4 ^-ica;") = mcx"->
dx
13.5
13.6
13.7
13.8
13.9
d , ^ ^ ^ V du . dv dw
da; ' dx dx dx
d ^ . du
(cw) = e-y-
dx
' dx
d , , dv ^ du
dx'
dx
d , , dw , dv . du
-;—{uVw) ~ UV-T' + UW-i r VVJ-j—
dx dx dx dx
d_fu
dx \v
v(du/dx) — u(dv/dx)
13.11 ^ = ^^ (Chain rule)
dx du dx
13.12 ^ =^ jV
dx dx/du
1313 — = ^^^^^
dx dx/du
53
DERIVATIVES
DERIVATIVES OF TRIGONOMETRIC AND INVERSE TRIGONOMETRIC FUNC
fi lA d . du
Ii».l4 ^— sinjt = cosM^—
dx ax
13.15 -T—cosu = — sinw-;—
dx dx
13.16 -J— tanw — sec^M^—
dx dx
13.17 3— COtU = — CSC^M-r-
dx dx
13.18 j--secM — secMtanu-j—
13.19 3— cscM = — cse M cot It -r-
dx dx
13.20 ^sin-i« =
1 du
13.21 -I— cos ' u =
dx
:-l,i —
13.22 Atan-.t ^ T^f^
(13: 1 + 1*2 f^x
~2< sin '«< 2
-T- [0 < cos J u < ■n\
-- < tan-i w < -
13.23 £eot-„ = ^1 10<cot-.«<.]
13.24 ^sec-.« =
du
±1 dw
13.25
dx
csc~^u =
\u\ Vm^-I t'* mVm2^ '^^
—1 dw q:l du
\u\ y/u^— 1
da;
tVw^ - 1
da:
+ if < sec-iM < jr/2
- if n-/2 < sec-^M < TT
- if < csc^u < tt/Z
+ if -7r/2 < csc-^u <
DERIVATIVES 01^
PanW
OGARIfHMIC FUNCTIONS
,**, d , ^ogg^ dtt
13.26 -3— log„M = -3-
dx u dx
1 3.27 -r- In M = -j- log. m = ,
dx dx ^^ u dx
1 du
13.28 :^a" = aMna^
da: rfa;
13.29 :^c" - e"^
da; da;
1 3.30 ^u^ =. A ew In « = e^^^^^[v\n u]
dx
dx
,du, , dv
dx dx
»IRIVATIVES of HYPERBOLIC AND INVERSE HYPERBOLIC FUNCTIOI
13.31 -^— sinh u
dx
, dv,
cosh u -J—
da;
1 3.34 -=- coth u —
dx
, ,, du
— csch^ M 3-
dx
13.32 :^coshM
da;
. , du
sinh u 3-
aa;
13.35 3- sechw =
dx
~ sech M tanh u 3—
da;
13.33 3— tanh «
da:
, , du
da;
13.36 -— cschM —
da;
— csch u coth tt 3—
dx
DERIVATIVES
55
13.37 Asinh-iM
ax
13.38 3— cosh~^M
dx
13.39 -^t&nh-^u
ax
13.40 ~coth-iu
dx
13.41 ^sech-iw
ax
13.42 -^csch-iu
da;
1 du
±1 dw
1 f^"
1 - m2 dK
1 du
l--u:idx
^1 du
uVT^^^^
-1 du
r+ if cosh-^ M > 0, u > 1"|
[- if cosh-iM< 0, M> ij
[-l<u< 1]
+1 dit
Imj y/TT^ ^ MvTT^ '^
[m > 1 or u < -1]
r- if sech-i w > 0, < u < 1"|
L+ if sech-i M < 0, < w < ij
[- if M > 0, + if u > 0]
HIGHER DERIVATIVES
The second, third and higher derivatives are defined as follows.
13.43 Second derivative = ±(^^ = ^^ = fix) = y"
13.44 Third derivative - ^(0) = ^ = /'"(*> = V
jr /jn— 1.,
13.45 nth derivative
- dx \dx^-i dx« ^ ^''* ^
= -. ^fl. ^'Va^.t-
k. irfiB iV •^^ -t-fc TTff T 'tf^^^^t'\— iT^T^^l^ WT ^^ .^^^
LEIBNITZ'S RULE FOR HIGHER DERIVATIVES OF PRODUCTS
Let Z?" stand for the operator 3—5- so that D'm = -^-5 = the pth derivative of u. Then
ax'^ ax
13.46
■where ( 1, ( J, -.- are the binomial coefficients [page 3].
As special cases we have
13.47
^ I \ d^ , „ dudv , d^u
13.48
dx^
d^v
du d^v . „ d^u dv . d?u
Jz—( \ — _u 9 . ^ 4- Q . A-
d«3
dat^ rfa; da;^ dx^ dx di^
DIFFERENTIALS
Let y = f{x) and ^y = f{x + &.x) - f{x). Then
13.49
Ax iz ^ dx
where e -* as ix -• 0. Thus
13.50 Ay 33 /'(x)Ax + *Ax
If we call Ax = dx the differential of x, then we define the differential of y to be
13.51 dy = f'{x)dx
56 DERIVATIVES
RULES FOR DIFFERENTIALS
The rules for differentials are exactly analogous to those for derivatives. As examples we observe that
'3-52 d{u±v±iv± ■••) = du :t dv ± dw ± ■■■
'3-53 d{uv) ~ udv + vdu
1 3.54 d f?i\ - ^^dw — udv
13.55 d{M") = nw-idu
' 3-56 d(sin u) = cos « du
'3.57 rf(cosM) = — sin udu
PARTIAL DERIVATIVES
Let fix, y) be a function of the two variables x and y. Then we define the partial derivative of f(x, y)
with respect to x, keeping y constant, to be
13.58 ^ ^ lim /(» + Ax, y) - fix, y)
dZ Al-*0 Ax
Similarly the partial derivative of fix,y) with respect to y, keeping x constant, is defined to be
13.59 il = li^ /(x, y + Aj/) - fix,v)
^y Ay-.0 Ay
Partial derivatives of higher order can be defined as follows.
13.60 ^ = A. f^] ^ ^ ±(^
dx^ dx \BxJ ' dy2 sy [dy
13.61 -^ = A/a/"\ j^ = ±f^
dxdy dx ydyj ' SyBx 3y \Sx
The results in 13.61 will be equal if the function and its partial derivatives are continuous, i.e. in such
case the order of differentiation makes no difference.
The differential of fix, y) is defined as
13.62 df = fdx + ^dy
ax dy
where dx = Aa; and dy = Ay.
Extension to functions of more than two variables are exactly analogous.
DEFINITION OF AN INDEFINITE INTEGRA1
li ~-= fix), then y is the function whose derivative is f(x) and is called the anti-derivative of /(ar)
dx " -^ /• dy
or the indefinite integral of f{x), denoted by | f{x)dx. Similarly if y - \ f{u) du, then ^ = f{u).
Since the derivative of a constant is zero, all indefinite integrals differ by an arbitrary constant.
For the definition of a definite integral, see page 94. The process of finding an integral is called
integration.
In the following, u, v, w are functions of x; a, b, p, q, n any constants, restricted if indicated;
e = 2.71828. . . is the natural base of logarithms; In u denotes the natural logarithm of u where it is assumed
that M > [in general, to extend formulas to cases where w < as well, replace In u by In |m|]; all angles
are in radians; all constants of integration are omitted but implied.
14.1
14.2
14.3
14.4
14.5
14.6
14.7
14.8
14.9
■ dx = ax
I af{x) dx = a \ f{x) dx
I (u±v±w:ii ■ ■ •)di£ = I udx ± I vdx ± \ w dx ±
i udv — uv — I vdu [Integration by parts]
For generalized integration by parts, see 14.48.
j f{ax) dx = — \ /(«) du
/w" + 1
Wdu = -^V-T» K^-1 [Forn = -l, see 14.8]
r — - In M if «> or In {-u) if u <
J u
= In ImI
du = e"
14.10 i a"-du = I e"'"
1 du
gU In a
In a
In a '
a> 0, a¥^l
57
INDEFINITE INTEGRALS
14.11
1 sin u du
=
— cos u
14.12
\ cos u du
=
sin u
14.13
1 tan u du
=
In sec u = — In cos u
14.14
1 cot u du
=
In sinu
14.15
\ sec u du
=
In (sec tt + tan «) — In tan ( o" + T
14.16
1 CSC u du
=
in (cBC u — cot u) — In tan —
2
14.17
1 sec^ u du
-
tanw
14.18
1 CSC- K du
^
— cot u
14.19
1 tan^ u du
=
tan u — u
14.20
1 cot^ It du
=
— cot M — M
_ . _, r . « . w sin 2« , , .
14.21 I sin^K du = o" 2 — — ^(w — sinu cosu)
1 4.22 j cos2 II du = — -\ 2 — — ^(m + sin u cos u)
14.23 Jsec»tan«du = secu
1 4.24 I CSC II cot u du = — CSC u
1 4.25 I sinh tt du = cosh u
14.26 I coah udu = sinhu
14.27 I tanhitdw — In cosh m
14.28 I coth It du = In sinh u
14.29 I sechttrfw. — sin-Mtanhw) or 2 tan"* e«
14.31 I sech- w d« — tanh u
14.32 j csch2 urfu = -cothii
x
14.30 i csch u du = In tanh ^ or — coth-'c"
14.33 I tanh^w du - u — tanhw
INDEFINITE INTEGRALS
59
14.34
14.35
14.36
14.37
14.38
14.39
14.40
14.41
14.42
14.43
14.44
14.45
14.46
14.47
14.48
I coth2 u du = u — coth u
I sinh^ u du = — r -^ — ^(sinh « cosh u — u)
1 cosh^ udu — \- — — ■|(sinh u cosh M + «)
I sech M tanh « du = — sech u
I csch u coth u du = — csch u
f du _ 1 _,M
J u2 + a2 ~ a ^" a
/du 1 - /u — a\ 1 ^. , " 9^5
-5 ^ = TT 1" ( . ~ ~-coth-i- m2 > o2
u^ — a?- 2a \u + aJ a a
/
/
^ oin 1 —
du
. '^" = In (m + Vw2 + a2 ) or sinh-i -
Vu2 + a2 "
r ^" ^ In {« + Vm2 - a2 )
— , — -sec~M-
wV^^23^ a I a
r dw ^ _l^^f a + ^u^ + a^
J uyju^ -t- a2 » " V tt ^
r du ^ _\^^{°:1S^^Z^
This is called generalized integration by parts.
^'''dx
IMPORTANT TRANSFORMATIONS
Often in practice an integral can be simplilied by using an appropriate transformation or substitution
and formula 14.6, page 57. The following list gives some transformations and their eflfects.
1 4.49 r F{ax +b)dx = - f F(«) du
1 4.50 r FcVax + b )dx = | J « ^(w) du
14.51 r Fi^fax + 6 ) dx = - f w'«-iF(M)du
14.52 f F(Va2 -x'^)dx = o J F(a cos w) cos «
14.53 j F(y/W+~^)dx ^ « ) F(a sec u) see2 «
du
du
where u = ax+h
where u = •\Jax+ 6
where u = v**3J + ^
where x = a sin u
where x — a tan u
1*
60
INDEFINITE INTEGRALS
14.54 J F{\Jx^ — a?- ) dx = a \ F{a tan u) sec u tan u du where x = a sec m
14.55
fFie^^)d. = If
F(u)
du
1 4.56 J F{ln x)dx - ( F(u) e« du
14.57 fFUin-i^j (te = orF{w)cosw
du
where u = e"^
where u = \nx
where u = sin^^ —
a
Similar results apply for other inverse trigonometric functions.
14.58 f Fisinx,co.x)dx - 2Jf(^^,{^\^ where u
= tang
Pages 60 through 93 provide a. table of integrals classified under special types. The remarks given on
page 57 apply here as well. It is assumed in all cases that division by zero is excluded.
14.59
14.60
14.61
14.62
14.63
14.64
14.65
14.66
14.67
14.68
14.69
14.70
14.71
INTEGRALS INVOLVING &x + b
7~r = - In (ax + b)
ax + a ^ '
Jxdx _ X b
r-r =: 5 In (ax + 6)
ax + b a a^ ^ '
f x^dx _ {ax + 6)2 2b(ax + 6) , 6^
I r-r ~ ~ — „ ■ ' i — 5 H ^ In (ace + 6)
J ax + b 2a3 a^ a^ '
/
/
/
^ (ax + &)3 _ Sb(ax + &)2 ^ BbHax + b) _ ^^^ ,^^ ■ ^
3a*
x^ dx
ax + b
dx _ i^ I w
x{ax + b) b \ax + b
dx _ 1 a^ / ax+b
zHax + b) ~ ~bx "^P^^l X
dx
x^{ax + 6)
dx
(ax + 6)2
dx
2ax-b o2
262^2 ^ 63 '" V ax + 6
-1
J (ax
/x'
(ax
a(ax + 6)
6
+ 6)2
2 da;
9, _,_ , , + — 5 In (ax + 6)
a^ax + 0) o2
ax + b
i3
62
+ 6)2 a3 a3(aa; + 6) a^
x^dx _ (ax + 6)2 Bb(ax + b)
26 , , , ,,
5 In (ax + 6)
(ax + 6)2
dx
J a;(o3; + 6)2
2a^
]
6 (ox
+
63
a*{ax + 6) a
+ ^ In {ax + 6)
! + 6) "^ 62^"(^aa: + 6J
a;2(ax + 6)2 62(aa:+6) 62a: 6^
1 , 2a , / ttic + 6
+ TTT In
INDEFINITE INTEGRALS
61
14.72
14.73
14.74
14.75
14.76
14.77
14.78
14.79
14.80
14.81
14.82
dx ^ {ax + 6)2 3a{ax + b)
xHax + 6)2 26*x2 "^ Mx
_ 3a^ , / ax + b
bHax + b) b* \
dx
-1
2((m; + 6)2
-1
aHax + 6) ^ 2aHax + 6)2
o3{ffla: + 6) 2aHax + 6)2 ^ "^3 '" ^'^^ + "'
362
63
a? a'^{ax + 6) 2a4(ox + 6)2 a
T In iax + 6)
a^x<
2aa 1 , / aa; + 6
263(a» + 6)2 63(a« + 6) 63 I x
S
X
/
/
/
J
J
J
J
If n ^ -1, -2, see 14.62, 14.67.
J
If 71 = -1, -2, -3, see 14.61, 14.68, 14.76.
r r-r- + — \ r^ I a!'"(aa; + 6)" i dx
m + m+1 wi + n + lj '
(ax + 6)3
xdx
{ax + 6)3
a;2(ja:
{<u; + 6)3
(ax + 6)3
dx
x{ax + 6)3
dx
xHax + 6)3
rfx
x3(ax + 6)3
(ax + 6)" dx =
_ — <t 2a 1 , 3a . f ax + 6
262(ax + 6)2 63(ax + 6) 63x 6* ^" \^ x
a-'x2 4a3x _ {ax + 6)g _ ^ ,„ / ax + b \
ax + 6)2 65(ax + 6) 265x2 b^ \ x )
265(
(ax + 6)" + i
(n+l)a ■
If n = -1, see 14.59.
x2(ax + 6)"dx - io-^ + bY*' _ 26(ax + 6)n-H2 62(ax+6)"-^'
(n+3)a3 (Ti + 2)a3 ^ (n + l)a3
14.83
J x'"(ax + 6)"dx
x"'(ax4-6n + * -mb T „ ,, , t. ■,
-^-^ , ,. - 7 — ; r-rr I «'"~Vax + 6)"dx
(m + n + l)a (m + m + l)a J '
:n+l)6 ^ (n + l)6 J "" (ax+ft) rfx
M.84 J
/
dx
14.86
yax + b
dx
■\/ax + 6
x2 dx
Vax+6
INTEGRALS INVOLVING y/axTh
Zyjax + 6
2(ax-26) / — TT
3a2 ^°^+^
2(3a2a:2 _ 4a6x + 862) _
-^ -—- -' Vox + 6
15a3
14.87
14.88
J
J
dx
xyax + 6
1 /v'ax+T — ■/&
■/& \v'ax+6 + /6,
2
tan
dx
x^ax + 6
[ax -
6a! 26 J
^/=:6
Vax + 6
6
dx
x-\/ax + 6
[See 14.87]
62
INDEFINITE INTEGRALS
14.89
14.90
14.91
14.92
14.93
14.94
14.95
14.96
14.97
14.98
14.99
14.100
14.101
14.102
14.103
14.104
ax + b dx =
I xVax + b dx —
I x^y/ax + b dx =
/" y/ax + b
r \/ax-
J ^
J' g"*
VcuF + 6
/
2V(ax + 6)3
3a
X
■\fax-¥h
dx =
dx =
dx =
2(15a2a;2 - 12abx + 862) .
2yiax + 6 + 6 f
dx
xvax+6
Vax + 6 a /• d«
2 J a:.
"^ -' ■^•\/ax+ 6
[See 14.87]
[See 14.87]
2x'«V«a: + 6 2m6 T a;*"- ^
a J
dz
rfa;
x^yax + 6
ax + 6 dx =
dx = -
dx =
\ x"'V
/■\/ax-¥h
/y/ax + b
x^
r (ax + 6)'"/2 dx
f x(ax + 6)'«/2(ia:
j xHax + 6)'n/2 dx
r^ax+_6r^
f (gg + b)"'^2
J
r dx
J x(ax + 6)"'/2
{2m + l)a {2m + l)a J ^^ + (,
Vaas + 6 _ (2wi - 3)a /" dx
*(m-l)6x'"-i {2m~2)6j a^^-iVo^Tfr
= ,o ^Tox (t^ + &P^' - /o ^^',^ r a;"*" Vttg + 6 dx
(27n + 3)a^ ' (2m+3)aJ
V ax + 6
s r
dx
(m-l)x'"-i 2(m-l) J a,m-iV^^T6
-(ax + 6)3/2 (2m - BW r Vox + 6
(m-Dbx""-! (2m
^ 2(ax + &)t'" + 2>/2
a(m + 2)
-5)a f
-2)6 J
X"!-!
dx
_ 2(ax+6)tw» + 4V2 ^ 26(ax+6)t'" + 2)/2
a2(m + 4) a2(m + 2)
2(ax + 6)t'"+g>^2 46(ax + 6)<'"+<)/2 262(aa; + 6)<"^+2J/2
a3(m + 6) aH7n + 4) a3{m+2)
dx =
dx =
2(ox + 6)'"/2
J
(aX + 6)<'n-2J/2
+ 6 I i;:::^-^^^^- — dx
m J X
_ (aa; + 6)t"'+2)/2 mo r (ax + 6)"'/a ^^
6x
2
(m - 2)6(ax + 6)('«-2>/2
26
+
IS
dx
x(ox + 6)<"'-2>/2
INTEGRAIS INVOLVING ax -\- h AND px + g
14.105 J
14.106 ^
14.107 f
14.108 r
dx
14.109
J
{ax
+ h){vx + g)
xdx
(ax
+ 6)(px + g)
dx
(ax
-f- 6)2(px + g)
xdx
(ax
+ 6)2(px + g)
X2dx
In
px + q
bp — aq \ ax+ 6
-r-^ i~ln(ax + 6) - ^ In (px + g)
6p — a^ a p
+
In
6p — aq 1 ax + 6 bp — aq V ox + 6
px + q
1
In
ax + 6
bp — aq 1 6p — aqr \px + q
6» 1
o(ax + 6)
+
(ox + 6)2(px + q) (bp - aq)aHax + 6) {bp - aq)^ \ p
^\n{px + q) +
^fc^hn(ax + 6)
INDEFINITE INTEGRALS
14.110
/
dx
-I
dx
(ax + h)^{px + q)" (n - l)(6p - aq) ]{ax + 6)'"^Hp3: + 9)"~^
+ a(m + n-2) J^^^^jj^^p^^^j^^,
14.111 1^ 7—dx = — + ^ „ In (px + q)
J px + q p p2
' rl J(°^+^)"'^' + in-m-2)a C -^^^^^dx]
{n-l){bp-aq)\{px + q)"-^ ' J (px + «)" ^ J
14.112
.) ipx + q)^
dx —
Zl__ji^±^ 4- mibp-aq) f
(n — 7n — l)p UpK + g)"^! ^
(px + 7)"
dx
(n-Dpltpx + g)"-! '"''J (px + g)"->'*^
INTEGRALS INVOLVING v/aa; + b AND px + q
14.113 f P^ + ^ dx = 2(apa' + 3ag-2M^/^^:^
-* ^/^.x + h 3a2
14.114
/
dx
'yjp(ax + b) — y/bp —
aq
In
(px + g) y/ax+ b
y/bp — ag Vp V vp("^ + 6) + \bp —
aq
\aq — bp vp
tan
-1 I p(oa; + 6)
"^ a? — 6p
14.115
/
\ax + b
px + q
dx =
2\ax + b vbp — aq lyJp{ax + 6) — yjbp — aq
+ =— In
P pVp \yjp{ax + 6) + y/bp — aq^
2yax + b 2y/aq — bp
pVp
tan
_, / p(qg -
"Y ag -
+_61
6p
2(px -r qy ■ 'y/ax-ro hp - aq f
ij>:c + qYy/ax + bdx = ^^^^fs)^ + (2« + 3)p J ^/^^^ip^
14.117
r dx_
-^ (px + q)"- V
2(px + g)« + JVax + 6 ^ 6p - ag r ( pg + ?)"
/ax + 6
(2n - 3)a T
-6p) J
dx = ,„ ■ „ -- h T^ - , ».'' ■ I 'T ' "" ■ da;
(px + g)-V^^Tt ~ Tn-l){aq-bp)(px + qY~^ ' 2{n - l)(aq - bp) J (pa; + g)"-> V^^+6
(2n + 3)p
Vax + b
^;i^ +
dx
lAllfi r <Pa^ + 9)" ^ _ 2(px + g)"Vgx+6 2n(ag-6p) r (px + g)"-idx
■ J ,/:r7XT ''"^ - {2n + l)a + (2^+1)^ J ,/— Tft
Vox+fe
Va^ + 6
(px + g)"
—\ax + b
V ax + 6
dx
,,o r V "3^ -^ o _ -V«'^'^P g r dx
**■"' J(px + a)" "^ ^ («-l)p(p:r + g)'«-i "^ 2(7i-l)pJ (p^ + ^j„-i^/^^:jrb
"~^^ ^ - ^^ UMI I il
INTEGRALS INVOLVING y/ax + b AND v^xTg
14.120
/
dx
■\/(ax + 6)(px + q)
- In (y/a(px 4- g) + y/p{ax + b) )
2
tan-i
'—ap
V —p(ax + 6)
a{px + g)
4^f
14.121 r
xdx _ V(aa: + b)(px -f g) fey +
ag
\/(ax + 6)(px + g) **P ^"P '^ ^/ (ax + b)(px + q)
J
dx
U-
64
INDEFINITE INTEGRALS
14.122
14.123
14.124
|V{a-+6){px+,)d. ^ '"p^ \l^ + "^ V(a. + &)(px + g) - ^''^g-;;^^' 1^
da;
V(aa: + b){px + g)
VX+ q . _ Vi'^ + &)fP=g + g) ^ gg-ftp
ax + b
dx
dx
^ 2a
2^/axTb
f
dx
y/{ax + h){px + q)
{px + q) V(aa; + 6)(pa: + g) (ag — 6p) Vpa; + q
14.125
J a:2 + tt2
14.126
C xdx
J ar2 + a2 "
14.127
C k2 dx
J x^ + a^ "
14.128
r x^dx
J x^ + a^ ~
14.129
r dx
J x{x^ + a^)
14.130
C dx
J ic2(a;2 + a2)
14.131
f dx
14.132
f dx
J {ic2 + o2)2 -
14.133
(* xdx
J {x2 + a2)2 "
14.134
r x^dx
J (x2 + a2)2 -
14.135
f x^dx
J (a;2 + a2)2 "
14.136
r dx
J ar(a2 + o2)a
14.137
r dx
J a;2(a;2 + o2)2
14.138
r dx
J ic3(x2 + a2)2
14.139
f dx
J (a:2 + a2)n
14.140
/• xdr
J (a:2 + a2)n
14.141
z' dx
J X(x2+a2)n
14.142
r ar"' rfa;
J (a;2 + 0.2)71
14.143
r da;
CEGRALS INVOLVING a;^ +
- tan-i —
a a
= iln{a;2 + a2)
X — a tan~^ —
a
y2 /,2
|--|ln(x2 + a2)
2a2^" 1x2+ a3
■ -5 i tan 1 —
2a2x2 2a4 ^"^ l^x^ + a^
a; + J^ tan-i-
2(x2 + a2)
— z , _1_ *an-i —
2(x2 + a2) 2a a
5^ + - In (x2 + a2)
2(x2 + a2) 2 ^ ^
1 + A.i„ ^ ^^
2aHx^ + o2) "^ 2ai \x^ + a^
-^- ^ - -3-tan-i-
a'fx 2a4{x2 + a^) 2a5 a
1_ _ 1 _ J, , / xM
Za*x^ 2aHx^ + a^) o* \^x2 + a2y
rfx
(x2 + a2)''-i
2(m-l)a2(ar2 + a2)"-i (2n
^1
2(ri-l)(x2 + a2)n-i
^ 1 + J^ (' dg
2(Ti-l)a2(x2+a2)n-i ^ a2j x(x2 + a2)"-i
r a;'"-2dx _ 2 r x'"-2dg
J (x2 + a2)n-i " J (x2 + a2)"
J_ r dx \_ C dx
0.2 J a.m(3;2 + a,2)n-i ^2 J a:">-2(a;2 + tt2)n
INDEFINITE INTEGRALS
65
14.144
14.145
14.146
14.147
14.148
14.149
14.150
14.151
14.152
14.153
14.154
14.155
14.156
14.157
14.158
14.159
14.160
14.161
14.162
r dx
J 3:2 - a2 -
■dx
- a?
1 , f x - a \ 1 . , , a:
2a \x + a J a a
pn(x^-a^)
I o, x — a
a: + ^ In ,
2 Va: + a
C x^dx
J a;2 _ „2 -
r x^dx
J a;2 — a2
/ dx
iB(a;2 — a2)
C dx _ _1_ , J_ , / x-a \
J xHx^ - a2) a^x "^ 2a3 \x + a/
^ x^{x^ — a2)
J (g2-a2)2
f gdg _
J a;2rfa; _
(g2 - a2)2
(x2-a2)2
y + yln{g2-«2)
1 , /g2_(i2
^ -J. In
2a2a:2 2a^ 1x2 - a2
- A: In / '^ - '^
-1
2{g2-a2)
+ -lln/=«-'^
2(a:2-o2) 4a \x + a
~^^ -i- 1 In {g2 - o2)
rfg
2{x2 - (i2) 2
-1 , 1
+ ^In
2a2{a;2 - qS) 2a4 Xx"^ - ti^
- T%ln
a^a; 2a4{x2 - a2) 4a5 \. g + a
+ 4ln
2a^g2 2a4(a;2 - a2) a« Kx-^ - a
-3B
2n-
J g{g2-a2)2
C dx
J g2(;c2-aa)2
r dx
J a;3(a;2_a2)2
r dg
J (g2-a2)n
/ g dg _
{g2 - a2)«
r dg ,
J g(g2 — a2)"
/ g"* dg _
(g2 - a2)n
r dg ^ 1 r dg \^ r
J g'"(g2-a2)n a2j a;'"-2(g2 - a2)n a2 J
2(Ti-l)a2(g2-a2)"-i (2n
-1
t-3 r dx
-2)o2j (a;2-a2)"-l
2(n-l)(g2-a2)"-i
-1 J, r dg
-I)a2(g2_a2)n-1 a2j x(g2-a2)"-i
2(n
J g'"-2dg 2 r g"'-2dg
(g2-a2)n-I ^ " J (g2-a2)n
dg
g'"(g2-a2)"-i
66
INDEFINITE INTEGRALS
14.163
14.164
14.165
14.166
14.167
14.168
14.169
14.170
14.171
14.172
14.173
14.174
14.175
14.176
14.177
14.178
14.179
14.180
14.181
J a^ — x^ 2a \a —
J a2-
or — tanh~ ' —
X
dx , a, I a + X
x^ 2 \ a — X
c^ dx
2 2
j^^ = -^-~}n(a^-x^
f dx ^ J_ / gg
J x(a2 - a;2) 2a2 '" \^a2 -
f dx ^ _J^ , J^, /« +
J xHa2 - x2) a2x 2o3 *" \^a -
f '^a: L- , J_ 1 f-^Lj]
J x3(a2-x2) 2a2:c2 ^ 20" \a2 - ara^
r dx ^ _
J (a2-x2)2 2a2(a2-x2) ' Aa^ "\a - x
J x dx _ 1
{a^-x
J (^
(a2 - a;2)Z 2(o2 - x2)
2dx
_ 1 l„/a+x
x2)2 2(a2 - x2) 4a V^a - x
J (a2 - -c2)2 2(a2 - x2) ^ 2 ^ '
+ :=^ln
r dx_
J x(a2 - x2)2 2a2(a2 - x2) ' 2a'' "* \^a2 - x^
/ dx _ — 1 , X , _3_ , I a + X
xHa^ - x2)2 a*x 2aHa^ - x2) 4a5 \a - x
r ^g ^ -1 , 1 + J_ In / g^
J x3(a.2 - x2)2 2a4x2 aa^ias - a;2) o* \a? - x2
f dx ^ X 2n - 3 r dx
J (a2-a;2)n 2(n- I)a2(a2 - x2)r.-i "^ {2n - 2)a2 J (a2_a;2)n-i
J xdx
{a2 - X
2)« 2(tt-l)(a2-x2)
/
dx
1 , i_ r dx
,-l)a2{a2 — a:2)n-» a2 J a:(a2-x2)"-i
x(a2 - x^)" 2(K
(a2-x2)n "• J (a2_3;2)n J (a^ - x2)n-\ : ■'< -
f dx ^ 1_ r dx , J_ C dx
J «™(a2-a;2)« a2 J a'n(a2- a:2)n-i "•" a? J 3:^-2(^2 _ a-2)n
■»- . r
r ' r>
1
INDEFINITE INTEGRALS
67
INTEGRALS INVOLVING V*^Ta^
14.182
14.183
14.184
14.185
14.186
14.187
14.188
14.189
14.190
14.191
14.192
14.193
14.194
14.195
14.196
14.197
14.198
14.199
14.200
14.201
14.202
dx
f . ""^ = In (a: + VicM^ ) or sinh " ^ -
cV«^ + o,^ a2
^ V^MT^ 3
xV'x2 + o2
ln(x + VkM^)
1 j a + y/x^ + d^
= - - In
a \ X
/- dx ^ Va=2 + a2
J a;2^a;2 + a2
a^x
Va;2 + o2 1 /a + \/x2Ta2
+ ^;-:i In
2a2a:2 2a3
3;V^2 + a5 0^2
+ ^ In (X + \/x2 + ^2 )
/ da: _
x3\/x2 + a2
I Vx2 + a2 dx =
rx2,/^2T^dx = .(x2 + a2)s;. _ g^vgrg _ 4 ^^ ^^ ^ ^^-.-^^
v' 4 8°
fxaV^T^d^ = (x2 + a2)5/3^^2(:,2 + ^2)3/2
./ 5 3
r y/x^ + a2 , „ „ ^ fa + y/x^ + a'^
J a
dx = y/x^ + a2 — a In
Vx^ + a2
J
./ X
dx - -
\/x2 + a2
+ !n (x + ^3:2 + a2 )
Vx2 + a2
dx = —
\/x2 + a2 1 /a + \/x2+a
2x2
--^In
J (x2 + a:
X dx —1
f x^dx _ —X
(x2 + a2)3/2 - .^/^5T^ + ln(x + -/^24^)
J (x2 + a2)3/2 ^^2 + a2
J x{x2 + a2
1 /a + Vi^Ta2^
;;ln
r dx ^ V^' + "■' X
J x2(3;2 + a2)3/2 a^X ^4y^2+^
r dx -1 3
J x3{x2 + o2)3/2
fa + y/^ + a2\
202x2^ + 02 2a4Vx2 + a2
+ 2!b1"
a + Vx2 + a2
68
INDEFINITE INTEGRALS
14.203 f (a:^ + a2)3/2 da: =
14.204 r x(a;2 + a2)3/2 da;
14.205 ra:2(x2 + a2)3/2da:
14.206 r a:3(3;2 + a2)3/2 da;
,4.207 I i:?i±^>^.x =
_ (^2 + a2)5/2
x(x2 + a2)5/2 a2a;{x2 + a2)3/2 a^^V^M^ „6 . , „ ,
- 6 24 16 - jglnCt + V^^T^)
_ {a:2 + a2)?/2 (l2(a;2 + ^2)5/2
J
(a;2 + a2)3/2
+ a2V»2 + a2 - a3 In
a + V^2T^
(x2 + a2)3/2
14.208 1 '-^ : ^ dx =
a;2
,4.209 I "'^ + f'"' d. =
(3;2 + ^2)3/2 3x-\/xM=^ g
2 + 2a2 1n(x + VxM^)
/
2:r2 +g/x2+a2--aln
INTEGRALS INVOLVING \/a;2 - a^
14.210
14.211
14.212
14.213
14.214
14.215
14.216
14.217
14.218
14.219
14.220
14.221
14.222
14.223
r^^^__ ^ In {x + y/x^ - a2 ), C -^^^ ^ V^^^^-
J ^a:2 - a2 J V^sTT^
f_x2d^_ x\/a;2-o2 ^g ^ ^
dx
= — sec~^
xy/x^ - a2 «
Va:2-a2 ^
xyx'^ — a2 ^2
J
r
•^ x^yjx"^ — a?-
C dx
•^ a; Va:- - a^
r Va:" - a2 dx = ^
jx^f^^^dx = M^!)!^
I x^yj x^ — d
.7 5 3
C Vx2 - a2
J ^2
C dx
J (a:2-a2)3/2
-^\n{x-¥ y/x^-a^)
a;(a;2 - ^2)3/2 a^x^f x"^ ~ a^ a*, / ^ r^ 2\
2 da; = — ^ ;; 1 r- In (a; + \x^ — a^ )
dx = -\/x^ — a2 — a sec~' —
I a I
Vx^ - o2
dx = + In (a; + Va;2 - a^ )
-\/x^ — a2 J
2a:2
X
2a I a I
a2-/a:2 - a2
INDEFINITE INTEGRALS
69
14.224
14.225
14.226
14.227
14.228
14.229
14.230
14.231
14.232
14.233
14.234
14.235
14.236
J x dx _ —1
r dx
C dx
} a:2(a;2 - a2)3/2
r dx
- -3 sec 1
Vx^ - gg _ a;
1 3
)3/2
J(.-„.j3.,, . x(.2 - „2)3. _ Sa^xV^ ^ 1^, ,^ ^^ ^ ^.^.^^
fxHx2-a^)^ndx = (a:^-a2F^^ a2(,2-„2)5/2
t/ 7 5
•^ ^ 3 I a I
r (x2 - a2) 3/2
J X^
(a;2_ 0.2)3/2 3a:\/a;2 - a2 3
-■|a2 1n{x + Va^ - a2 )
(g2 _ q2)3/2 3^x2 - gg _ 3
2x2 "^ 2 2
INTEGRALS INVOLVING V^^TT^
14.237 r , '^^ = sin-i^
14.238
J v/q,2 „ 3.2
14.240
14.241
14.242
14.243
/~2 j_ x-Ja^ — x^ „2
Va" - a;2 2 2a
r ^'^tfx _ {a2 _ ^2)3/2 — 5
r rfx ^ _]_ / g + Va2_a.2
J xVa2 - x2 '^ " V 3:
Va^~x2
J 3;Z^tt2 _ a.2 a^x
X rfx
x3Va2 — x2
Vo^ - x2 1 /a + Va^-x
7^ In
2a23;2 2a3
fr
r
70 INDEFINITE INTEGRALS
r Va2 - x^
14.244 I yfd^^x^dx = ^^^^— ^ + f sin-'l
14.245 I x^a^-x^dx = _ (5i__^!l!^
I xy/a~ — x^
14.246 J a;2^a2-x2dx = _^i^_-!_ + v_ +__sj„-i_
14.247
14.248
J O
14.249 I- — s dx = sin-'T
14.250 J ^^ — ^ — rfa: = - n^g + ;r: 1° ' ^
2ic2 ^ 2a
^^•^5^ /(^ '^
2^x2)3/2 „2V^rr:i^
14.252 Tt^'^'^
)^'^ V^
14.253 I Y^y 2^372" = , - s'n^-
,4.254 f-V^i^ = V^^:^+^^=
/dx 11, f g + Va^ — a?2 '\
p dx _ -1 3 3_ J / tt + Va
14.257 J ^3(„2_«;2)3/2 - 2a2x2V^3^^ ^ 2a^^^^^^ ^a^ "V
14.258 J (a2-x2)3/2da: = ^ ^ ^ + ^^8 + ga* sin »-
J/-,2 _ 3:2^5/2
X(a2-x2)3/2dx = -IB-^J—
,4.260 J .^(.-.=)«.. = -2(2!:^ + -'-<"'--''"' ^ ^^^f^ + fi^in-.f
14.261 /.»(.-x=)«<£x = i^-=fg^- °^'°^ --''"'
,4.264 Ji °^-f'" .. = _(£!^^_?VHEE?+|„„(£±3^)
INDEFINITE INTEGRALS
71
14.265
J
dx
ax^ -f 6a; + c
2 , 2aa- + &
tan~'
V4ac — 62 yf^a^^^^
In '
V62 - 4ac \ 2aa; + 6 + \/b^ - 4ac ,
If 62 = Aae, ax^+hx-¥c = a{x + 6/2a)2 and the results on pages 60-61 can be used. If 6 - use
results on page 64. If a or c =^ use results on pages 60-61.
14.266
dx
ax~ + hx ■¥ c
'^"''^ J aa;2+6x + c (jn-l)a a J ax^ + 6x + c a J
dx
ax^ 4- 6x + c
x""^^ dx
14.269
J
dx
= i^In
14.270
x(fflx2 + 6x + c) 2c \^ox2 + 6x -I- c
dx 6
2c J
ax2 + 6x + c
dx
ax^ + 6x -I- c
KL
dx
az2 + ba; + c
C dx _ _6_, / ax2 + &x + c N _ J_ . ^^ - ^ac T
J x2(ax2 + 6x + c) 2c2 \ x^ ) ex %<^ J
,.„, f dx ^ _ 1 b f dx ^ f —
J x"(ax2 4- bx + c) (n-l)cx"-i cj x'»->(aa;2 + bx + c) cj x"
Id 979 r tfx ^ 2ax + 6 , 2a. f
i*i.i/x J (^3.2 + ^3.4.^)2 (4ac - b2)(ax2 + 6x + c) 4ac - b\}
dx '
-2{ax2 + 6x + c)
dx
14.273
14.274
14.275
/
(ax2 + bx + c)2 (4ac - b2)(ax2 + 6x + c) 4ac - b^ J ax^ + 6x + c
xdx __ bx + 2c b r dx
(ax2 + bx + c)2
2dx
. to
S • -
-bsj
r ^
J (ax2 +
J (ax2 + bx + c)"
{4ac - b2)(ax2 + fcx + c) 4ac
(b2-2ac)x + be + 2c T
aic2 + bx + C
dx
+ bx + c)2 a(4ac - b2)(ax2 + bx + c) 4ac - b^ J ax^ + bx + c
mdx -''"-' (m— l)c i x'"~2dx
(2n - m - l)a(ax2 + bx + c)
(n — wt)b C x"*~* dx
(m-l)c f
"-I (2n — m-l)aj
(ox2 + 6x4- c)"
.X
14.276
14.277 . j(^2 _j_ bx + c)2 ~ 2c(ox2 + bx + c)
14 978 r ^^ ^ 1 _3g r
1.^/ o J ^2(a^2 + 6a; + c)2 cx(ax2 + 6x 4- c^ c J
14.279 f
{2n-m-l)aJ {ax2 + bx + c)"
2'>-3 dx _ _ ^ f
x2''-2 dx
X x^^-^ dx _ 1 C x2"-3 dx _ c r X
(ax2+bx + c)'' ~ a) {ax2 + 6x + c)"-i a J (ax2
c dx ^ 1 _ J. r '^'^ I ^ f
J x(ax2+bx + c)2 2c(ox2+bx + c) 2c J (aa;2+6x + c)2 cj x(ax2 4-bx + c)
+ bx 4- c)n a J (ax2 + bx 4- c)"
dx . 1 r dx
dx
c2(ax2 + bx + c)2
dx
cx(ax2 + 6x 4- c)
1
(Ox2 + bx + c)2
_ 2b f
C J X
dx
(ax2 + bx + c)2
(m + 2to — 3)
x'"(ax2 + bx + c)" (wi - l)cx"'~Hax^ + bx + c)"-i (m- l)c
dx
dx
2(aar2 + bx 4- c)"
(m + n — 2)b
(m — l)c
J ar"*-'
(ax2 + bx 4- c)"
INDEFINITE INTEGRALS
INTEGRALS INVOtVING \/a^Tb^Tc
In the following resulte if 62 = 4ac, Vax^ + bx + c = yfa{x^ 6/2a) and the results on pages 60-61 can
be used. If 6 = use the results on pages 67-70. If a = or c - use the results on pages 61-62.
14.280
/
dx
14.281 r
14.282 r
yjax"^ + hx + c
xdx
■\/ax^ -f- 6a; -f c
x^dx
y/ax^ +bx + c
~^\Tl{2^/a^J ax^ + bx + c + 2ax + 6)
y/a
1 .„^i / 2ax-¥ b
L V-a \yt2-4acj V^
yjax^ -\-bx + c h
or sinh 1
y/iac — b^
2oJ
dx
y/ax^ + bx + e
_ 2ax - 36 j „ 2 , , , , 362- ^^c f
da;
Voic^ -i- 6a; -|- c
14.283
14.284
14.285
/
dx
xy/ax- -I- 6a; + c
dx
_ J_ In ( 2y/cy/ax^ +bx + c + bx + 2c
yfc
sin'
6a; + 2c
■■^y/ax^+JxTZ
I \^ax^ + bx + c dx =
y~^ \\x\yjb^-4ac
_ y/ax^ +bx + c b
2c J ^
1 • U-. / bx + 2c
or — -— sinh ' /
Vc yx\y/4ac-b^
dx
y/ax'^ + bx + c
{Zax + 6) Va«2+"6xTc 4ac — 6^
4a
+
8a
/
dx
y/ax^ + 6a: -f c
14.286 rxVax2 + 6a: + cda: = (ax^ + 6x -H ^)3/2 _ b{2ax + 6) . ^^, ^ ^^ ^ ^
_ 6(4ac - 62) r da;
1^«' J y/ax^+bx + c
14.287 JWart^-hfrx + cdx = ^"'^'J^ ^^^^ + ^'^ + '^>''^^ + ^^'leJ^^ ' f V «=^^ + 6a; + c da;
dz
14.288
14.289
/y/ax^ + bx + c , , h f
^ dx = yjax"^ + 6a; -I- c -h | j
di
-I- c
J
Vaa;2 + &g-l-c , _ yfa^^^hxVc
yjax^ -H 6a; -I- c "^ xyfaifi^Vbx + c
•^ a;i
da; — —
14.290 /^
14.291 r
dx
2(2ax -!- 6)
+
-/
da;
\/a«2 -i- 6a; -f- c ^ -^ xy^zM^TxTc
/
dx
ax2 + 6x -H c)3/2 ^^^^ _ j^,^ y^2+6^ + ^
xdz 2(6a; + 2c)
14 292 r ^^'^'^ :^ (26^-4ac)a; + 26c ]. r da;
J („^2 + bx -K c)3/2 ^(4^^ _ ^,^ Vaa;2+6x + c "^ a J ^ ^^2 + 5^ + ,
14.293
J x(
dx
-I-
^ (62 — 4ac) vtta;2 -I- 6x -I- c
_ 1 r dx b_ r dx
iax^+bx + cY'^ ^^^2 + 6^ + ^ "^ cj ^^/^^FT6^T. ^cj (ax^ + 6a;
14294 f ^^^ - o3;2 + 26x -t- c 62 - 2ac r dx
J x2(ax2+6x-Fc)3/2 fl23:^ax2-K6x-Fc 2c2 J (a;^2 + 6^,4^)3/2
36 r dx
^''^'^ x\/ax2T6xTc
14.295 ( {ax-^-^bx + c)^ + ^'^dx = (2ax + 6)(gx2 + 6x-Hc)"+i/2
-^ 4a(m + 1)
+ c)3/2
INDEFINITE INTEGRALS
73
14.296
14.297
14.298
J,
2(2ax + 6)
(0x2 + 6a; + cY + 1'2 (2m - l)(4ac - 62)(ax2 + 6a: + cY~^'^
8o(n — 1) r da:
+
/.
dx
(2w-l)(4ac
1
-62) J (aa:
(ax2 + 6x + c)" + >'2 (2n - l)c(aa;2 + 6x + c^-^i^
dx
+
'J
x(ax2+6a; + c)"-i/2
2 + 6x + c)"-i'2
2c J
d:e
(oar2 + 6x + c)«+i/2
INTEGRALS INVOLVING x^ + a^
Note that for formulas involving x^ — a^ replace a by —a.
,4.299 f^^ = ^,„_i£±^ + ^tan-^^
1 , a:2 _ aa, + a2 j _, 2x-a
■T— In — -. — ; — 7K 1 — z tan ' p—
6a (x + a)2 ^^ „^
14.301 (' ff^, = |ln(a:3 + a3)
J 3;3 + a3 3
14.300
/ xdx _
a;3 + a3
dx
ay o u-v "
;.3 + „3) ,4.302 /^(^ = 3^3ln(^
j_ _ j_ x2 - Qx + g'i _ ^_ ^^^„, 2x^1^
a3x 6a4'" (x + a)2 ^,^ ^^3
ax + a.2
a:2 — fflx + g2
3a4\/3
14.303 J a:2(x3+a3) ^03; oa- vx -r or
wsAA r_-^5— - g , J_, (g + «)^
14.304 J (j.3 + a3)2 - 3a3(x3 + a3) "^ 9a5 ^" x2 - ax +
lAtn^ r ardx _ a:2 , 1 .^ a:2^axjt^
I4.J03 J (a:3 + a3)2 " 3a3(x3 + a3) ^ 18a* (x + a)2
,--*- r x2dx _ 1
J (x3 + a3)2 3{x3 + a3)
^^•^°^ J x(x3 + a3)2 ^ 3a3(x3 + a3) "*■ 3^ ^" Vx^ + a3 j
I^.JUlf J ^2(a^ + q3)2 a^x 3a«(x3 + a^) Sa^ J x^ + a-
14.309 J ^^.p^ = ^^ - "^ J -^T^
J^ —\\ C dx
x"(x3 + a3) = a3(n-l)xn-i " a V x"-3(x3 + aS)
INTEGRALS INVoK^^^^^C?
^ , 2x — a
-^tan-i 3-
Sa^VS aV3
2x -
+ tan"!
[See 14.300]
14.311
14.312
14.313
14.314
x2 + axV2 + o2
a^ + a* ~ 4a3\/2 "' V*^ " «*V2 + ©2
X
C xdx
} x^->r a'^
C x^dx
J X* + a*
J x^dx
x^ + a*
In
= ::-;; tan » -5-
2a2
1 /x2 — 0x^2 + a2
In
4av^ \a2 + ax\/2 + a^
1 . _. axv2
-tan 1 „ „
2a3V2 * " '^
^ tan- ^1^
2ay2 a:2 - a^
J In (x* + a*)
74
INDEFINITE INTEGRALS
14.315
/
dx
^ T^ In
x(x'* + a") 4a4 \x^ + a^
2aG^""'f^
14317 r ^'^ - ^_}: 1
14.318 f-r^ = J_ln/^^^VA
14.3.9 J,!^ = i^I-nfe
M.3ao J,5!i?|^ = -L,„
^ „, axy/ 2
^ tan ' -s 5
V2 * ~ **
tan-i-
a
z — a\ , 1 . _, X
— ; — 4- ;;- tan 1 —
x + aj 2a a
14.324 r
dx
1 , 1 , / 3:2 - a2
+ -r^ In
x3(3ri - a*) 2a'»x2 4a6 y ^2 4. ^z
INTEGRALS INVOLVING a;" ± a"
14.325
14.326
14.327
14.328
14.329
14.330
14.331
14.332
14.333
14.334
r dx_
J x{x^ +
+ a")
^ In ^^
na'" a;" + a"
fx"^*dx 1
/pwi— n (£2;
/ x"'dx _ r x"*~"da: _ „ T
(x" + a")' ~ J (x" + a")'-! "■" J (x" + a")*-
X dx _ J_ r dx J_ r <ix
x'"(x" + a")'' a*^} x'"{3;" + a")''-* a" J
dx
x"*'~"{x" + a^y
_ 1 / ^/x" + g" - \/a" \
xV^*^" + a" mV'a" \\/x*» + a" 4- yf^J
/ dx _ _1_ - / g" — a" \
x(x" — g") Tia" I X** /
Jxn-irfx 1, ,
—;, ;: = - In (x" — a")
X" — a" n ^ '
/ x^dx _ ^ r x"'-"<?x r x""" dx
{x^'-a^y * J (x"-g")'' J (x''-a")'— 1
r dx ^ 1 r dx ?^ f —
J dx
da;
2 . / o"
:cos^
INDEFINITE INTEGRALS
75
14.335
2m
,2m -p
(2fc - l)pir / a: + a COS [(2fc - l)W2m]
2 sin — s — tan^i
fc=i
a sin [{2k - l)jr/2m]
y
where < p ^ 2m.
2ma2m
1.2m -p j^^j^ 2m \ 2m J
14.336
a2m
fepr
m
k^
2 cos— ^ In [ a;2 — 2(ixcos-^ + a
14.337
14.338
1 '"■^' . fcprr . _, /a: — a cos (fc^/m)
o„-n 2 sin -^— tan ^ : — ., , -: —
^a2m-p ^ ^ 1 a sin (fcirArt)
where < p ^ 2m.
J x2m + 1 4- a2m + 1
+ 2^i^t^°^^"''^ + (-l)Mn(x + a)}
2(_i)P-i ™ . 2fcp^ , _i / x + g COS [2kirl{2m + 1)]
(2m+l)a2--P + ifc^i "'" 2m + 1 ^^"
(_l)P-i
2kpir
asin[2fc7r/(2m+l)]
2fe7r
-^ 2 cos^^^ In x2 + 2ax cos^^^ + a^
(2m+l)a2'"-p+ifc=i 2m+l \^ 2m + 1
(-l)p-Mn(x + a)
(2m+l)a2'n-p+»
where < p ^ 2m + 1.
j.2m + l _ (i2m+l
- -2 ^ _in ^'^/^ tan-i f ^^
(2m+ l)a^'n-P+i ^-f , ="" 2m + 1 1, a sin [2K/(2m + 1)]
_2 ■'• ZkpTT - /''^ ~ a cos [2fcjr/(2m + 1)}
+
(2m + 1)
1 ^ 2fcps- , / 5 „ 2fc:r , „
^ TT 2 cos -= — —■ In a2 _ 2ax cos ■= t—t + or
j(j2m-p+i^"^j 2m + 1 Y 2m +1
+
In {x — a)
(2m + l)a2m-p + i
where < p ^ 2m + 1.
INTEGRALS INVOLVING sin ax
i> ■
14.339
14^40
14.341
sin ax dx — —
sin ax dx =
cos gjc
a
sinour
a2
X cos ox
2x f 2 x^
x^ aiaax dx = -j sin ax +( -3 — — J cos ax
14.342 (n^ainaxdx = (^ " ^) s'"*^ + fe" ^
14.343
14.344
14.345
14.346
14.347
/^
sm ax , (ax)3 , (ax)^
dx = ax — ' '. +
/ sin.
a:'
J sin
J sir
ax
dx = -
3-3!
sin ax
+
X
»J
5'5!
cos ax
X
dx [see 14.373]
ax
dx
— — In (esc ax — cot ox) = - In tan -^
a a £.
1 j , (ax)3 , 7(ax)5
2(22n-l - l)B„(ax)2n+l
■^ (2n+l)! ^
sin^ ax dx — -^^ —
X sin 2ax
4a
^ M
^':
-S.
INDEFINITE INTEGRALS
14.348
14.349
14.350
14.351
14.352
14.353
14.354
14.355
14.356
14.357
14.358
14.359
14.360
X sin2 ax dx = ^ - ^ ^^" ^"^ _ ^os ^aa;
4 ia 8a2
sm^axdx = -^osax ^co^ax
a 3a
sin^ axdx - ^ - s'" ^"^ , sin Aax
8 4a ^ 32a
dx
/
/
/
J
.f
J
J 1 — sin
J
J
sin^ ax
rfa;
sin^ax
— cot ax
a
2a sin2 aa: 2a
+ t:- In tan -r-
sin P.C sin 9X dx = ^'" ^P " g>^ _ S'" ^P + ?)»
2(p-9) 2(p-h9)
[If p = ±q, see 14.368.]
^ ltan(f+^
ax a V4 2
i — sin ax a \ 4 2
4lnsin -?-
dx ^ -ltan(^-°^
1 4- sin ax o \ 4 2
J 1 + sin
{X — sin ax)2
r ^
J {X - si
= -^tan(^-«-^
sin ax a V 4 2
dx
_i_ ^ 1 . / f ox
H s In sin T + ^r-
„ -tan ^ + ^
2a \ 4 2
+ f- tan3 [ 5 + ^
6a \ 4 2
(l-sinax)2 - "2^*^"(4"'2";- 6^**"H4-T
J
dx
p -f g sin ax
a-y/^2Tr^
tan-i
p tan ^ax + q
Vp2l-g2
In
p tan iax + 9 — vg^ — p2
14.361
14.362
14.363
14.364
14.365
14.366
14.367
14.368
If p= ±q see 14.354 and 14.356.
(?x o cos ax . JJ i^' dx
J
Q COS ax
+
(p + 9 sin ax)2 a(p3 - g2)(p ^ ^ gin ax) P^
If p = ^q see X4.358 and 14.359.
+ q sin aa:
Vp^ +
\/p2 _ q2 tan
/dx 1.1 y'p2 4- q,2 tan ax
/
f^rmsinaxdx = ^ ^^ cos ax ^ mx-"-isinax _ m(m-l) C ^rn-2 ^j^^^ g^
J a a^ a^ J
rsinox^^ = _ sinaa^ _^_ _g_ T gosox ^^ [see 14.395]
f sin" axdx = _sin"-iaxcosax ^ »^i_l T ^.^„_,
J an n J
p^ + q'^ sin2 ax
p^ — (p- sin2 ax
Vp^^2 P
^(72 _ p2 t;an ax + p
lapyjrp- — p2 Vvg^ — p^ tan ox — p
' 2 ax dx
/ dx _ — cos gx , n — 2 ^ dx
sin" ax a(n — 1) sin"~* ax n— 1 J sin""^^^^.
J xdx _ — z cos ax 1 w — 2 f" x dx
sin"ax a(n — 1) sin""" ^ ax a2(w,- l)(n — 2) sin"-2ax n — 1 J sin"^^
INDEFINITE INTEGRALS
77
^OTEGRAIS rNVOlVlWs''^^^^
14.369
14.370
14.371
14.372
14.373
14.374
14.375
14.376
14.377
14.378
14.379
14.380
14.381
14.382
14.383
14.384
14.385
14.386
14.387
14.368
14.389
/, _ sin ox
cos ax dx —
a
/, _ cos ax , X sin ax
X cos ax ax — — +
a^ a
;2 COS ax dx — —^ COS ax + [^ — -^ ) sin ax
a a-
jx^cwaxdx = (^-±jcosax + [^-—Jsmax
Xcosax . , {ax)2 (ax)* {ax)'^ 4. . . .
CS21^dx = -S£l^^ a (^^^^^dx [See 14.3431
J X^ X J X
= "In (sec ax + tan ax) = — In tan t + "^
COS ax
J
xdx ^ J_ Uax)^ (ax)^ 5(oa;)a
cos ax
+i5r+
"^ (2n + 2)(27i)! ■*"
/„ , a; , sin 2aa;
cos^ ax dx = -x +
4a
J- _, x^ , ;c sin 2ax , cos 2ax
C , , sin ax sin^ ax
I cos3 ax dx — r
J a 3a
JZx sin 2ox , sin iax
cos* ax dx - ^ + -^^ + -^2^
J
_ tan ax
cos^ ax
dx
r_^ = sinax + 1 intan('^ + ^
-/ cos3 ax 2a cos^ ax 2a \4 2
J, sin (a - p)x , sin ( a + p)x
cos ax cos px dx = ^ : f-
2(a-p)
2{o + p)
Xr^
dx
1 .ax
= cot -r-
cos ax a ^
h
St
dx
cos ax
dx
X . ax , 2 , . ax
- cot -r- + -5 In sin -^
a 2 a^ 2
1 ^ ax
- - tan -^
+ cos ax a 2
Jo-
/
_JEA^_ ^ ^tan^+ ^.Incosf
+ cos ax a 2 a^ £
dx 1 ax 1 3 ax
dx
(1 + cos ax)^
1 ^ ax , 1 . „ ax
■^ tan -^ + -F- tan-f -^
2a 2 6a 2
[If a ^ ±p, see 14.377.
78
INDEFINITE INTEGRALS
14.390
14.391
14.392
14.393
14.394
14.395
14.396
14.397
14.398
h
dx
q cos ax
a\p^ —
. tan- ^ ^/ (p - q)/{p + g) tan^a;
In
tan ^ax + v'(g'+p)7(9--p)
[If p = :tg see
14.384 and 14.386.1
dx
J {p + q cos ax)^
dx
ay/q^ — p2 ^tan ^ax — y/{q + p)li.q — p)l
q sin ax _ p C dx
J
a(g2 _ p2J(p ^ q, (.pg g^jg^
, p tan ox
tan"'
+ g cos ax
[If p = ±g see
14.388 and 14.389.
p2 + g2 cos2 aa: apy/^^T^^ y/p^ + q^
J p^ — q2 cos^ ax
dx
1 . 1 P tan ax
■ — tan~i ~
apVp^ — q^ yjp^ ~- g2
p tan ax — yjq^ — p^
2ap-\/q^ — p2 \p tan ox + ^/q^ — p^
In
x-" cos ax dx = = — »ui_ux ^ »«x — cos ax s — ■ I x"" ^ cos ax dx
a a^ o^ ./ _
Tcos^ ^ _^osax _^ C^^dx [Seel4.365]
fcosnaxdx = sin ax COS"-' ax ^ ZL^li C
J an tf- J
r do
J cos"
sm ax
ax
xdx
a(n — 1) cos'
X sin ax
^ « - 2 r da
"^ ax n — 1 ^ cos"-
cos""2aa! dx
dx
cos" ax a(re - i) cos"" ' ax aHn — l)(n — 2) cos''-^ oa;
+
K-2 r xd
n — 1 J cos"*-
dx
2 ax
INTEGRAIS INVOLVrNG sin ax AND cos ax
14.399
14.400
14.401
14.402
az cos ax dx =
J sin
fsinpxcosgxdx = - COs(p-q)x _ cos {p + q)x
J 2{p - q) 2(p + g)
I sin"
I cos"
ax cos ax dx =
2a
sin" * ^ ax
{n + l)a
[If n = -1, see 14.440.]
ax sin ax dx = —
cQs"'^^ ax
(n+ l)a
[If n = -1. see 14.429.]
1 4.403 1 sin^ ax cos^ ax dx = — —
14.404 f- —
J sm a
8 32a
dx
14.405 f^-
J sm
14.406 r^-
J sm
X cos ax a
dx
= - In tan ax
1 , ^ / V , ax \ 1
= - In tan t + -h" ~ r
2 ax cos ax a \i 2 / a sin ax
dx
1 . ^ ax , 1
= i In tan -r- +
14.407
/ii
ax cos^ ox a 2 a cos ax
dx 2 cot2ax
sin^ ax cos^ ax
a
>-^
INDEFINITE INTEGRALS
79
14.408
14.409
14.410
14.411
14.412
14.413
14.414
14.415
14.416
14.417
14.418
14.419
14.420
COS ax
dx = -
, 1 , , I ax , IT
J cos
sii
I
dx ~
dx
a a 2
cos ax(l ± sin ax)
dx
sinofl:{l ± cos ax)
r dx
.) sin ax ± cos ax
ax dx
f sin aa
sin ax ±
J
cos ax
cos ax dx
1 + f lntan(^+J
2a{l ± sin ox) 2a \2 4
1 , 1 , . ax
+ ;;- In tan -r-
2a(l±cosax> 2a 2
av^ V2 8
X 1 , , . j_ .
-^ ^ ^ In (sin ax ± cos ax)
sin ax =: cos ax
sin ax dx
X 1
= ±— + — In (sin ax ± cos ax)
J p4
I = In {p + g cos ax)
+ q cos ax aq '
cos ox dx
J p + g sin ox aq
C sin ax dx
In (p + g sin ax)
1
(p + g cos ax)" aqi^n — l)(p + g cos Ox)»~ '
C COS ax rfx _ —1
(p + g sin oz)" ag(m - l)(p + g sin ax)"-i
.7 p sin
./ p si
dx
ax+ g cos ax a^jrP■+q^
In tan
ox + tan ' (g/p) \
dx
sin ox + g cos ax + r
a-\]i^ ~ p2 - g2
tan"
^ / p + (r — g) tan (ax/2)
In
p _ Vp2 + q2 _ r2 + (r - g) tan (ax/2) \
a\/p2 + ^2 _ r2 "" Vp + Vp2 + g2 - r2 4- (r - g) tan (ox/2) /
If r = g see 14.421. If r2 = p2 -j- ,2 ggg 14.422.
14.421
14.422
14.423
14.424
14.425
/
dx 1 , / , , ax
■• T — TT-T T = — In g + p tan ^
p sin ox + g(l + cos ax) a-p \ 2
dx
p sin ax ■\- q cos ax ± vp^ + g* aVp^ + g^ V
-1 ^J.^ ax^tan-Mg/p) \
J p- SI
dx
_ 1 , _.^ I p tan ox
sin^ ox + g2 cos^ ax apq V g
r dx
J p^ sin^ ox — g^
In
cos- ax 2opg \ p tan ax + q
p tan ax — q
J==
sin"" ax cos" ox dx —
sin"* ' ^ ax cos" "^ ' ax , m
a{m+n) m +
sin"" + ^ ax cos" "" ' ax
o(m + n)
+
/3in"''"2 ax cos" ax dx
Jsin"^ ax cob''"^ qx dx
80
INDEFINITE INTEGRALS
J C(
14.426 I ?H^^dx
cos" ax
a(n
sin'"~^aa; _ m — 1 C sin*" ^ ax
— l)cos"~^aa; n — lj cos''~^c[x
(i(re — 1) cos
— sin"* ^ aa;
a{m — n) cos"~i oa;
1 gg _ wi — n + 2 r _^
)s"~^ ax n — 1 J cos
sin"" ax
co&"^^ax
dx
+
m — 1 I s]n"'~"2 ax
cos" ax
- 1 Tsi
— M J I
da;
14.427
/CDS'" aa; ,
^ *^ac =
sin" ax
— CDS'"" ^ ax _ m — 1 r cQS*"~^ ax
a{n — 1) sin"~i ax n — 1 J sin^~^ ax
— cos'" + ^ax m — 71 + 2 C cos'" ax
a(m— 1) sin"-' ax n-1 J sin"-2ax *
— 1 r cos"
— n J sii
COS"" ' ax
,a(m — m) sin"- 'ax wt
cos"* -^ ax
sin" ox
14.428
/ii
dx
sin"" ax COS" ax
J^ ^ m + 7i- 2 f
"• ax cos"-' ax n — 1 J si
dx
a{n — 1) sin"
^1 _j_ m + ti-2 r
a(«i — 1) sin™-' ax cos"-' ox m— 1 J sin'"-
sin"" ax cos"--' ax
dx
2 ax cos" ax
IGRALS INVOLVING tai
14.429 I tanoarcZx = Incoaax = -Insecax
J « a
14.430 ftonZaajd* ^ t^EM _ ^
14.431 r tan* ax dx = ^^^ + ^ In cos ax
J 2a a,
14.432 rtan"a«sec2axdx - t&n^^^ax
J (7i+l)a
J ti
14.433 I I^Si^dx = ^ In tan ax
tan ax a
1 4.434 I 17:7^::: = r In sin ox
tan ax a
M.435 J.tan^.. = ^ 1^ , <^ , ?^ , . . . ,
22"(22n-l)B„{ax)2" + i
(271 + 1) !
+
14.436 J^dx = ax + Iggli + g<^ + ■■ ■ + ''"<'!" " ^^^.^^'^f""^ +
9 ' 7B
(27t - 1)(2tc) !
14.437 fxtanSaxdx = ^-^^2^^ + ^ In cos ax - -^
14.438
/
dx
px
+ ., , — 57 In (.Q sin ax + p cos ax)
p + 9 tan ax p2 + ^2 ^ ^(p^ + q^)
14.439 ftan-axdx = tan"-iax _ C^j^n-ZaxOx
J (n — l)a J
INDEFINITE INTEGRALS
81
INTEGRALS INVOLVING cot ax
14.440 r
14.441 r
cotaxdx ~ — In sin ax
a
cot^azdx = -i^tox _ ^
a
14.442
14.443
J
j cot"
X
14.445 r
cot3 ax da = - S°^.BE - 1 in sin ax
fix C8c2 oa; dx = —
2a a
cot" + ' ax
{n + l)a
14.444 I — dx = In cot a*
cot ax a
dx 1 ,
— 7 = In COS ax
cot ox a
^AAAJ^ C * J iJ (*«)^ (ax)"
14.446 I X cot ax az = -^<ax — ^—~ Tor"
14.447
/
cot ax , 1 £Ei _ (ttx)^
X ax 3 135
22«F„(ax)2''-
(2n + l)!
22«F„(ax)2"->
(2n-l)(2ji)!
14.448 fxcot^oxdx = -?LS^^ + ^insinax -^
J o a* 2
dx „ pg q
14.449
J
p + g cot oar p2 + g2 o(p2 -|- g2j
14.450 rcot«axdx = - cot"-' ax _ rcot"-2axdx
J (n - l)a J
In (p sin ax + q cos ax)
INTEGRALS INVOLVING sec as;
14.451 I secaxdx = - In (sec ax 4- tan ox) = — In tan (-—+--
J a ^ o V 2 4
J
14.452
sec2 ax dx =
tan ax
14.453 fsecaoxdx = 5e£«fLtanax + 1 j^ (ggc ax + tan ax)
J 2a 2a
14.454 fsec-oo^tanaxdx = ^^Sl°±
J na
14 455 f_^5_ ^ sin ax
J sec ax a
r ,_. _ 1 J(ax)2 , (ax)* , 5(ox)8
/
14.456
sec ax dx =
a2 I 2 "^ 8 "^144 (2n+2)(2Ti)!
■iMJi^-w I sec ax J , ^ (ax)2 ^ 5(ax)^ ^ 61 (ox)" ^ , gt.(ag)'"' ^
14.457 I dx = inx + '-^ + -^ + -^^^ + ■•■ + ^^^^^ +
X
14.458 I X aec2 ax dx = - tan ax -\ s In cos ax
J a a2
82
INDEFINITE INTEGRALS
+ q COS ax
14.459 r ^ = a_P f-
J q + p sec ax q q J P
14.460 (sec- ax dx = aeC-'^ax tanax ^ n-2 f ^^^^^ax dx
J a{n — l) n — 1 J
'*■»
INTEGRALS INVOLVING cac ax
14.461 I CSC axdx = - In (esc ax - cot ax) = - In tan ^
14.462 fcsc2aa:cte = -S^ax_
J a
14.4«3 /csc=a... = _ esc .. cot „. ^ ^ ,„ ^„ ^
14.464 f CSC" ax cot ox dx = -Hc^L«£
J na
14.465 f— ^5_ - _ COS ax
J CSC ax a
(ttx)3 , 7_(ox)5 , , 2{22"-i-l)B„(ax)2'> + i
■'.! ••»
1 4.466 J X esc ax dx = ^ ■{ ax + ^-7^ + 'i i ^L 4- • ■ • +
18 1800
(2n + 1) !
+
J
14.467 I H^i^dx = _JL + ^+ 7(^_j^ ,,_ ^ ^
2{22n-i-l)F„(ax)2"-i
ax 6 1080
14.468 I X csc2 ax dx = —
14.469
(2n-l)(2n)r
_ X cot ax , 1 ,
~ H 5 In am ax
a a'
J dx — * _ p r
q -\- p CSC ax ^ 9 J
dx
P + g sin ax
[See 14.360]
14.470 fcsCoxdx = _cscn ^axcotax ^ n-2 fcsC-^axdx
^ a(n — 1) n — 1 J
INTEGRALS INVOLVING INVERSE TRIGONOMETRIC FUNCTIONS
14.471 fsin-'-dx = x sin-> - + yJa^-x^
J a a ^
I X sin ' -
J a
14.472
''^ - 'T-TJ"^" 'a + 4
14.4/3 I x2 sin-i - dx = -r-sm ' — I ;;
J a Z a V
C sin-Mic/o) rf_ - « . (x/a)3 1 » 3(x/a)S 1 • 3 ■ 6(x/a)7 ,
J X a 2-3-3'''2-4-5-5^2'4-6*7*7
f sin-'i
J xa
14.476 r^sin-i^j dx = x ( sin-i ^ j - 2a! + 2^52^7^ gi^-i £
14.474
14.475
Jx/a) J _ _sm
in-i j x/a) _ 1 f a + y/a^-x^
a
-f
INDEFINITE INTEGRALS
83
14.477
14.478
14.479
14.480
14.481
14.482
14.483
14.484
14.485
14.486
14.487
14.488
14.489
14.490
14.491
14.492
/
J COS"'— da; =
a
z cos~^ - dx
a
\ x^cos-^-dx
a/ a
T cos"' (z/g)
J X
J cos~' {xia)
x^
lecos"^- — yJa^ - x^
a
- 1 £ _ x-\/a^ — x^
a 4
3 a 9
dx
X
dx
fin. -J
_ cos~' (x/a)
sin~i (x/a) ,
dx
X
[See 14.474]
dx
( tan-i-
J a
\ a:tan-> -
_ cos-'(g/g) + 1 In / « + A^a^ - x^ \
X a \ X J
a:fcos-i-j - 2a: - 2y/a^~x^ eos"'-
ztan-i- ^ ^In(x2+a2)
da: = ^(x2 +
o2) tan-i ^ - ^
^ X _ (z/a)3 (a:/a)5 _ (z/g)!
a 32 "^ 52 72 + •■•
a; a 2a \^ a;2 y
xcot-i I + |ln(x2 + a2i
rtan-Ma:/a)^^
./ X
J tan-Mx/a) ^^
Jcot^'— dx =
a
/x cot~^ -dx =
a
Jx2 cot~i -dx =
g
Jcot-Mx/g)^
^)
-i{x2 + a2}cot-i^ + ^
X3
|ln^- J tan-Mx/a) ^ fS,e 14.486]
_ _ cot-i(x/g) , 1 , fx^ + a^
~ X + 2^^" -^2-
14.493 j sec-'- dx =
X sec-i a In (x + \/«^ ~ «'
X sec-' - + g In (x + V^^ ~ "'
) <
'x2-a2)
-1 * ^ ^
sec ^ — < n
o 2
n < sec ' - < n-
<i a
14.494 i xsec-' * dx
a/ a
.-. X
, X ^ IT
sec-' — < 7;
g 2
<
^ < sec-' —
2 a
<
14.495
/
x^ sec - ' — dx =
•. 1
x3 a: gxVx^ — a2 g^ , , , /-^ s.
y ^^ a~ — 6 ~ y In (a: + V^^-a^) <
^^ ■=««-! ^ _i. a* Vx2 — a2 , a^ /—
T^'''^ a + 6 "^ yln{x + Va^2_„2)
(in '
I > '
■]•.
-1 ^
sec ' -
a
^'i
n < sec ^ —
84
INDEFINITE INTEGRALS
14.496 f^^^Af^da: = ^ In
J X 2
14.497 r^ec Hxia) ^^
J X-
14.498 fcse-i-da; =
14.499 f a; CSC- 1 -da:
2'""^ + a: + 2.3.3 + 2.4.5.5 + 2T4T
sec-"(3;/a) , \/a;2 — o^ „ ^ .a:
< sec-i —
1 • 3(a/x)5 1.3. 5(a/x)?
2.4.5-5 "•" 2.4.6-7-7
(a;/a) ^ \/a;2 - pg
■ ax
a<i
sec"' (x/g) _ yf:
X
x^ ~ a2
ax
X CSC"' - + a In (a: + y/x^ — a^
X cac-^ a In (x + \/^
a:* _, a: , aV^^TT^
2 a 2
a:^ , a: aylx"^ — a^
— CSC - ^ — i
.3 a 2
■a^]
IT < sec-' - < IT
2 a
< csc-i- < ^
a 2
-^ < cse-i - <
1 a
< csc-i - < ^
a 2
n < CSC-' - <
2 a
14.500 r a;2 CSC-' -da;
J a
14.501 { esc Mx/g) ^^
-=- CSC ^ h — - —
3 g 6
x^
" ■ ^In(x + Va:2-g2)
^ + -.n
/ CSC 1 (g/g)
K^ , X axyjx^ — a2 a,3 ^
y"^^"'a- 6 -|ln(x + V^^^)
I a
<
<
-1 * ^ iT
CSC ' - < -
a 2
o < csc-1 - <
£■ a
<x
^ (g/a;)3 1 . 3(g/a;)5 1 • 3 • 5(g/g)7 , \
2*3.3 "^ 2.4.5-5 2-4.6.7.7 *"/
14.502 I £^£lM^rf^
/
csc~' (a;/g) _ Va:^ — o-
3^ ga:
<
14.503
1 a:"" sin-* - dx
14.504
\ x"^ COS"* —da;
J a
_ g'»+^
a;m + l
J
14.505 I x^t&n-^-dx
C5c-^(xla) yjx"^ — a^
X ax
a:"' + i - _, ic 1 C -^ - ,
sm 1 7-rr I dx
m + 1 a m+ IJ ^^2 - x^
— -— - cos ' — b r^ I — — dx
m + 1 a m + ij^/^^--^
x^ + '^ . ^. X a C* K^ + i
— - tan I — - I -5— — ^ da;
m + 1 a m + lt'ai^ + a^
, a; . ;r
CSC-* — < 77
a 2
^ < csc-J- <
2 a
14.506 f a;- cot-i ^dx = -^ cot"' ^ + -^ ( 4^ dx
J a m+1 gm+ljx2 + g2
14.507 I z^ sec-i^da:
14.508 j a;"'csc-''-da:
a:"' + ' sec-' (g/g) _ a C x*" dx
./;
m + 1 «t + 1 J y^^zr„2
J z"' da;
\/a;2 - a2
a;m + i 3ec-i (a;/i
sec-' (a;/o) _| a
m+1 m +
a;*""*"* CSC-* (xla)
* CSC-* (x/a) , g_
m+1 m +
- r^
'da;
\/a2l^
a-m + i
esc ' (x/g)
m + 1
g r a;'" dx
m+1 J y^^IT^
< sec-'-
0^2
o < sec ' —
d a
<
- < V
a
<
CSC-' - < 5
a 2
K < cse-1 - <
2 a
INDEFINITE INTEGRALS
85
IGRALS INVOLVING
14.509
14.510
J
J
xe'^dx = — (a:--
a \ a
14.511 r»2««d^ = '—(^2^^+^
14.512 { x-efl^dx = ^:i^ ~ ^ C x^-ie<^
J a o. J
dx
= ^f ^n ^ ■»«""' , n(w-l)x"-2
(-l)"rt!
a"
14.513
14.514
14.515
14.516
14.517
14.518
14.519
14.520
14.521
14.522
14.523
14.524
^ ^ 1-1! ^ 2-2! ^ 3-3!
f — d* - -^°' + ^ f ^"^ ^,
r rfa; X I
J dx _ * I 1 1
dx
pC^ + qe"'^
— — tan->( -/£e<"
aVP7 \ V 9
In
if n = positive integer
. 2ay—pq \e^ + yl—qlv>
{ ^?,mhxdx = ^'^i'^^iribx-bcoBbx)
r e" cos bx dx = ^(a cos bx + b sin &a;)
J a2 + 62
r a:e<" sin bx dx = '^^'"(o sin bx - b cos ba) _ c<"{(o2 - 62) sin 6g - 2ab cos 6g}
J a2 + 62 („2 _,_ 52)2
f xe<^ cos bx dx = ^^'^^'^ '^"^ ^^^ + & s'" &a^) _ e^ICa" - 6^) cos 6x + 2ab sin 6a;}
J a2 + 62 (a2 + 62)2
r«-l„xd. = e'^lnx _1 fe^^^
J a a J X
J e- sin" 6a. do: = £!^^in^(„ sinfcx - n6 cos6«) + ^^^^^ J e- sin-2 ft^ d^
j'6-cos«6a:d« = ^^ll^^^^^f^bx ^ nb ^r.hx) ^ "^^^^^ ^ e'^ co^^-ibx dx
INDEFINITE INTEGRALS
INTEGRALS INVOLVING In a:
14.525
14.526
14.527
14.528
14.529
14.530
14.531
14.532
14.533
14.534
14.535
14.536
14.537
14.538
14.539
dx = a; In X — :b
Jin.
J ^2
x\nx dx — -z-{lnx — ^)
Xaim+i / 1 \
X- Inxdx = ;j^^^p^ (^In X - -^j^j^ j [If m = -l see 14.528.]
J X 2
Jinx j^ _ _lil£ _ 1
x2 X a;
j ln2 a; dx = x In^ a; — 2a; In x + 2a;
J in" X dx _ In" + ^ x
n+ 1
[If n = -l see 14.532.]
f-T^ = In(lnx)
J X In X ^ '
r^ = ln(lnx) + lnx + ill^ + il^+ ■■■
Jinx 2-2!3'3!
Jinx 2*2! 3'3!
I In^xdx = xln^ic — w | ln"~ia:!dx
\ x^Xn^xdx = ^"'"^^^""'^ ^ rx'«ln"^ixdx
If m = -1 see 14.531.
j ln(x2 + o2)dx = xln(x2 + a2) - 2x + 2atan-i-
rin(x2-o2)dx = « In (x2 - o2) - 2x + o In (|-^
rx"»ln(x2^a^)dx = x-^Mn(x^^a^)„ 2 C ^^^
J ' m + l wi+lJx2±o2
INTEGRALS INVOLVING sinh aa;
14.540
J sin
h ax dx =
cosh ax
14.541 fxsinhaxdx = xcoshox _ sinha£
J a a^
14.542 I x2 sinh ax dx = I 1 — g ) cosh ax — s ainh ax
INDEFINITE INTEGRALS
87
14.543
14.544
14.545
14.546
14.547
14.548
14.549
14.550
14.551
14.552
14.553
14.554
14.555
14.556
14.557
14.558
14.559
14.560
14.561
J X 3-3! 5-5!
/sinHox , _ sinh ax , T cosh ax , ._ , , „^„,
— ^2 — "* - ~ — "*" " J — X — f 14.565]
J.
x2 "- a:
dx _ 1 , , , OiB
-r — - In tanh -^r
nnax a 2
dx
sinhax a^] 18 1800
sinh^ ax dx
2a 2
_ X sinh 2ax cosh 2ax x^
dx _ coth ax
J sin
rsinh2oxdx = sinh ax cosh ax x
J 2a 2
Jxsi
J sinh2ax ~ ^
r sinh ax sinh px dx = sinh (a + p)x _ sinh (a - p)x
^ 2{a + p) 2(a - p)
For a = ±p see 14.547.
(' sinh ax sin px dx = « cosh ax sin px - p sinh ax cos px
•^ a-2 + p2
r sinh ax cos px dx = ° cosh ax cos px + p sinh ax sin px
J a2 + p2
2(-l)''(23"-l)gJax)2"+i
"^ (2n + l)! "^
dx
In
qe
_ *" + p - Vp2 + q2
+ 9 sinh ax aV^TT^ "' V^e^^ + p + ^/^+^
dx — q cosh ax _i. P
(p + g sinh ax)2
+
h
dx
/
da:
a(p2 + g2)(p + g sinh ax) p^ + q^ J p + q sinh ax
^ tan-i Vq^ - p^ tanh ax
ap\j(^ — p2 P
p2 + g2 smh2 aa;
p + Vp2 — gg tanh ax
, 2apv'p^ — g2 \^p — \/p2 — qr2 tanh ax
In
/^
dx
p + \/p2 + g2 tanh ax
2 - g2 sinh2 ax 2apyf^^^^ \p - V^^TT^ tanh ax,
J x*" sinh ax dx = x*" cosh ax _ m T j^_i cogj^a. j^j. [ggg 14.585]
J„- i,„„, j„ _ sinh"^i ax coshax n — 1 f . ,,_„ ,
sinh" ax ax — I sinh'' 2 ^x dx
TsinJ^^^ = -sinh ax ^ _g_ f £2^ ^^ [See 14.587]
J X" (m — 1)3!"- > n — 1 i^ x" 1 L J
/ dx — — cosh ttx _ w — 2 r dx
sinh" ax a(n — 1) ainh"^^ ax n — 1 J sinh""^^^;
/ xdx _ — X cosh ax 1 n — 2 r x d;
sinh^ax a(n — 1) sinh"-' ax a2(m— l)(n — 2) sinh''~2ax n — ij ainh"-
X dx
8»
INDEFINITE INTEGRALS
INTEGRALS INVOLVING cosh aa;
14.562
14.563
14.564
14.565
14.566
14.567
14.568
14.569
14.570
14.571
14.572
14.573
14.574
14.575
14.576
14.577
14.578
14.579
14.580
14.581
14.582
i_ J _ sinh ax
cosh ax ax =
X cosh ax dx —
a
X sinh ax
cosh ax
x^ cosh ax dx — —
cosh ax
2x cosh ax
+ ( — + ^ ) sinh ax
X
cosh ax
x^
d, = in^ + i£^+4«^+i^ +
2*2!
dx = -SS^l^ +
4-4! ' 6*6!
sinh ax
"J^
dx [See 14.543]
cosh ax a
xdx _ 1 J(ag)
cosh ax
(gg)^ , 5(ax)'>
8 "^ 144 "^
, „ , X , sinh ax cosh ax
cosh^ ax dx = -r- + ^
, „ , x2 a; sinh2ax
X cosh^ ax dx = -r + :
4 4a
cosh 2ag
8a2
dx
tanh ax
cosh^ ox
J
J
X
/
J
/
J
J
J
/
J
/, a sinh ax sin px — p co sh ax cospx
cosh ax sin px dx = 2 , n
I cosh ax cos px dx =
/
J
/
J
J
J (cosh ax — 1)'^
/ dx .
p + 17 cosh ax
/ dx
{p + Q cosh ax)2
(-l)"B„(ax)2« + 2
"^ (2n + 2){27i)! ''■
t. J _ sinh a-p)x . 3mh(o+p)a!
cosh ax cosh px dx = — -rr-^ ^^ 1 n- , „■■ —
2(a — p) 2(a + p)
_ a sinh ax cos px + p cosh ox sin px
a2 + p2
dx
cosh ax + 1
dx
cosh ax — 1
xdx
cosh ax + 1
X dx
cosh ax — 1
dx
(cosh ax + 1)^
dx
1 . , ax
= — tanh-Tp
a 2
1 ., ax
= coth -^
a 2
X , ax 2 , , ax
= - tanh ^ - -5 In cosh -5-
a 2 a^ 6
= 2^*^"*^T~6^'^"*' 2
^ J-coth^-icoth3^
2a
6a
2 ^ _, Qe'^ + P
In
ge'
or + p — '\Jj^ — q^
q sinh ax P r_
a(q2 - p2)(p + g cosh ax) q^ _ p2 J p
dx
+ g cosh ax
INDEFINITE INTEGRALS
89
14.583
J P'-
dx
14.584
14.585
14.586
14.587
14.588
14.589
h
(p- cosh^ ax
dx
In
/ p tanh gg + Vp^ - q^ \
Vp tanh ax — Vp^ ~ 9^/
2 + ^2 cosh^ ax
2apy/f^ — q2 \^p tanh ax — Vp^ ~ 9^
—1 . p tanh ax
- rn-n^J
1
V^23^
p tanh ax + Vp^ + g^
2apVp^ + ^ \p tanh ax - y/p^ + q^.
In
1
Vp2 +
tan-1
j x"" cosh ax dx =
J
ap
x"" sinh ax
p tanh ox
\/p2 + g2
_ m /*
cosh- ax dx = cosh"-'axsinhax ^
"-^J
x"-' ainhox dx [See 14.557]
cosh" ^2 da; dx
Tcoshox^ ^ -cosh ax _a_ Tsinhox ^ [See 14.559]
f dx _ sinh ax n — 2 T da
cosh" ax a(n— 1) cosh""' ax m— 1 J cosh"
dx
/
cosh" ax
X dx
:.flf r.
^',. t.
X sinh ax
+
cosh" ax a{n — 1) cosh"" ' ox (n — l)(n — 2)a2 cosh'»~2 ^x
n-2 r xdx
n — 1 J c03h"~2 ax
INTEGRALS INVOLVING sinh ax AND cosh aa;
14.590
14.591
14.592
14.593
14.594
14.595
14.596
14.597
14.598
14.599
14.600
14.601
sinh ax cosh ax dx =
sinh^ ax
2a
J. . V J _ cosh (p + tf)x , cosh (p — g)x
ainh px cosh qx dx = — -/^^ ^ + — 57^^= — p-
2(p + g) 2(p - g)
C sinh" ox cosh ox dx = 8mh" + 'ax ,jj ^ ^ _j see 14.615.]
. . n LD J _ sinh 4ax x
sinh^ ox cosh^ ax dx = — -rr — -z
32o o
dx
= - In tanh ax
sinh ax cosh ax
^^ = -^Uin-^Binhax-S^h^
sinh^ ax cosh ox a a
r cosh" ax sinh ox dx = cosh" + Ux [if n = -1, see 14.604.]
J (n + l)o ^
J
J sii
J
/ CO
si
J
dx
sinh ax cosh^ ax
dx
sinh^ ax cosh^ ax
55^*1^ + ilntanh^
o a /
_ 2 coth 2ax
a
sinh^ax , sinh ax 1 . _, ._. ._
— : dx = tan ' smh ax
cosh ax a a
=.25h!°£dx = ^2^^^^ + I In tanh ^
sinh ax o a ,2
dx
cosh ax (1 + sinh ax)
1 j„ /I + sinh ax \ + 1 ^^^_, ^^
Za V cosh ax I a
I'.' ■
J. '
90
INDEFINITE INTEGRALS
14.602
14.603
J si
dx __ J^ ]j^ tanYi — + ^
sinh ax (cosh ax + 1) 2a 2 2a(cosh aa; + 1)
dx
14.604
14.605
14.606
14.607
14.608
14.609
14.610
14.611
14.612
14.613
14.614
J
1_ . ^^ _L a£. _ 1
sinh ax (cosh ax — 1) 2a 2 2a(cosh aa: — 1)
INTEGRALS INVOLVING tanh ax
lanh ax dx = — In cosh ax
a
tanh2 ax dx = x —
tanh aa;
a
( t^nh^axdx = ilncoshax - ^^"^'^^
2a
f tanh" ax sech2 oa; da: = *^"^''V.°''
./ (n 4- l)a
X
J
sech^ ax J _ 1 ■ . .
; aa: = - In tanh aa;
tanh ax a
—. = - In sinh ax
tanh ax a
J
r tanh
J
* V . 1 J(ax)^ ((ia:)= , 2(ax)7
tanhaxdx = ^^1^-^+^
2 3
xtanh^axdx = ^ - ?-l5n»L«^ + X in cosh ax
(-l)n-122«(22n - l)g„(gx)2n + l
(27t+l)!
+
ax , (ax)3 , 2(ax)''
-dx = ax--y-+^5-
dx
px
p + qf tanh ox P^ — 9^ a(p2 — q^)
(-l)"->22'"(22n-l)B„(ax)2''-i
(2n-l)(2Ti)!
In (q sinh ax + p cosh ax)
+
14.615
14.616
14.617
14.618
14.619
14.620
r tanh" ax dx = tanh" ^ ax ^ T ^^^y^n-z ax dx
J a(n-l) J
^^^^^^ ^NTlolBn^NVOmN^otra
/
J
coth ax dx = - In sinh ax
a
coth^ ax dx = x —
coth ax
Jcoth»axdx = ilnsinhox - ^°*^^''''=
2a
rcoth«axc8ch8oxdx = -S^^^lll^
J (n + l)a
csch^ ax . _ 1 , ..
— dx = In coth ax
coth ox a
— ^ — = - In cosh ax
coth ax a
INDEFINITE INTEGRALS
91
14.621 I X coth ax dx
14.622 C X coth2aa;dx
\\ax + ^^-^^^
(_l)n-122nBJax)2" + l
225
(2n + 1) !
+
_ X X coth ax
+ — 5 In sinh ax
14.623
14.624
{2n-l)(2TO)!
C coth ax ^^ ^ _ ± + ^ _ (gg)^ .... (-l)"22"B„(ga;)2''-i
J X ax 3 135 "* (2n -lV2-n.il
+
+ g coth aa;
14.625 rcoth''a«dx =
p2-qz - a(p2 - ^a) ^" (^ »'"*» az + g cosh ax)
coth"~* ax
ain — 1}
J
+ I coth" •'^ ax dx
^^^^^^^^^^^^tAtS mvOLVfNG sechaa:
1 4.626 I sech ax dx = — tan~ ' e°^
J a
14.627 fsechaaxdx = ^"^°'
J a
14.628 Tsech^axdx = sechax tanhax ^ 1 ^_
./ 2a 2a
14.629 fsech^oxtanhazda; = _ sech^L"^
•^ na
14.630 f —S^^— - sinh gg
J sech az a
14.631 fxsechaxdx = A W _ W^. , S(ax)6 (-l)«g„(axpn.2
^ a2 [ 2 8 ^ 144 ^ (27i + 2)(2m)!
+
14.632 r X sech2 axdx = ^^"^"^ - ^ in cosh ax
14.633 fsechoa ^^ ^ ^^^ _ (^)2 Siax)-* _ 61(ax)« (-l)"g^(ax)2"
J SB 4 96 4320 "^ 2n(2K)! "'"
14.634 r ^ ^ ^_P f dx fS*.pi4KRii
J 9 + P sech ax g Q J p + q cosh ax ^ ^ ^^'^^^^
14.635 fsech" axdx = sechn-2 ax tanh ax ^ ^^ r ^^^^„,, ^^
•-' a(n — li ■» — II
INTEGRALS INVOLVING
14.636
14.637
csch ax dx = - In tanh —
a 2
csch^ ax dx = —
coth ax
14.638 r cschs azdx =^ - csch ax coth ax _ ^ i ^^ , ax
J 2a 2a 2
1 4.639 f csch" ox coth ax dx = - ^s^^" '^^
f' na
INDEFINITE INTEGRALS
14.640
J
'^^ = -cosh arc
csch ax
,4.*4, ..csch»x^ = -l„._<^+?^ +
14.642
14.643
14.644
14.645
X csch2 axdx = ~ £cothax ^ 1 j^ ^j^j^ ^^
/
/
J csch Q
r
J g + P
_1^n/')2n-l_
+
2(-l)''(2
l)B„(aa;)2"+i
(2m + 1) !
+
oar 6 "^ 1080
= 5. _ p r
*7 9 J
da;
csch ax Q 9 -' p + g sinh aa;
— csch" ~ 2 ax coth aa;
(-l)"2{22'«-i - l)fi„(ax)2«-i
(2«-l){2n)!
[See 14.553]
+
csch" ax dx =
a(n-l)
_ « -2 r
n - 1 J
csch" ~ 2 ax dx
InTEoBS? INVoVviN^mVERSEHYPERBOUcflulmlOl
14.646
14.647
14.648
14.649
/sinh~^ —
a
da: =
X sinh~' vxM-a2
a
X sinh~* — dx
a
f x2 sinh-» - d3B
t+ti-''-*t-
a: scVk^ + a*
= — sinh"' — h
X ^ (2a2 - a:2) Vx^T^
J
sinh~' (a;/o) j^
*
? _ <g/'>)^ , 1 ' 3(x/a)5 _ 1 ' 3 ■ 5{x/ay , ...
2'3'3 2*4'5'5 2*4'6'7*7
a
ln2 {2z/a)
2
(g/x)^ , 1 ' 3(a/x)<
2-2-2 2*4'4-4
1 ■ 3 ■ 5(g/g)»
2*4-6-6'6
+
ln2 (-2g/a) (a/x)a _ 1 ■ 3(g/a!)< 1 ■ 3 ' 5(a/x)« _
2-2-2 2'4'4'4 "*" 2 -4 -6 '"6 '6
14.650 fsinh-Mx/g)^^ ^ _
14.651
14.652
14.653
14.654
14.655
14.656
14.657
14.658
J
;
X
J
sinh~' (z/a) _ 1 1_
a+ \Ax2 + o2
cosh 1 - da; =
a
X cosh"^ — dx
a
flc2 cosh""' -dx
a
X a V X
xcoBh-'(x/o) - Va;2 - o2 , cosh-Ma:/a) >
r co3h-'(x/a) ^^
+ if cosh-' (x/a) >
[ X cosh-' (a:/a) + -^x^ - a^ , cosh"' (x/a) <
ri(2x2-a2) cosh-' (x/a) - ^xVar^ - o2 , coah-'(x/a) >
[i(2x2-a2) cosh-' (x/a) + ^xy/x^ - a^ , cosh"' (x/a) <
r^x3 cosh-' (x/a) - ^(x2 + 2a2) yfx^ - aZ . cosh- Hx/a) >
1^x3 cosh-' (x/a) + ^(x8 + 2a2) ^x^ - a^ . cosh"' (x/a) <
= ^[|ln2(2x/a) + i^^+4:4i^ + UA4<^ +
2.2-2
0, - if cosh-' (x/a) <
2.4-4-4 2-4.6-6-6
\x\<a
x> a
X < —a
cosh~' (x/a) ^ _ _
tanh-' -dx =
a
r cosh-'
J
J
J
_ cosh-' (x/a) _ 1, / a + Vx2+aA h if cosh-' (x/a) > 0,
" '" ' ' + if cosh-' (x/a) < 0]
X tanh-i - dx
a
xtonh-'- + ^In(a2-x2)
^ + i(x2-o2)tanh-»f
x^ tanh-' — dx
a
= f + ?'-''-| + f-(-"^-^")
INDEFINITE INTEGRALS
14.659
14.660
14.661
14.662
tanh-' (x/a)
dx =
— ^
a
+ f^/°)^ , (x/a)^
tanh-i jx/a)
J
/
J coth-i ^dx = z coth-1 X + I In (ar2 - a2)
X ^ 2a " \ a2 - a;2
a:^
coth-i - dx =
14.663 f x2 coth-i ^ da;
14.664 poth-Mar/g) ^^
14.665 f coth-i (ar/g) ^^
J x^
'' a
_ ax^ , x^ .,, X , a^
32
52
+
14.666
14.667
14.668
14.669
/
sech-i -dx =
/^
sech-i^dx
a
f
sech-i (x/a)
dx =
csch-i -dx =
a
.-1 X
1 4.670 ( X csch - 1 - rfa; =
J a
14.671
r csch~^ (g,
/g)
dx
14.672 C xrn ainh-^ ^ dx
J a
_ cQth- '(ar/tt) , 1 ,
a; sech-i (a;/g) + o sin-' («/«), sech->(3;/a) >
a: sech^Mx/g) - a sm~^ [x/a), sech->(ar/g) <
J^9;2sech-Ma;/a) - laVa^ - x^, sech-Hx/a) >
[ ^x2 sech-i (z/o) + ^gVa2 - ^2, sech-Ma;/a) <
^ 2.2-2 2.4-4M
i In {a/x) In (4a/x) + i^^^ + ^ ' ^(x/g)^ ,
I." 2*2-2 ^ 2M'4-4
« cach-i ^ ± a sinh-i | [+ if a: > 0, - if x < OJ
T "^^"^^ ^ - ~^ [+ if X > 0. - if X < 0]
' ^ In (x/g) In (4g/x) + li^l^ _ 1 ' 3(x/a)^ ^ _ _
2'2'2 2»4'4'4
^ 2-2.2 ^ 2-4-4-4
_g ^ (g/x)3 _ 1 ■ 3(a/g)5
a; 2*3'3 2'4"5'5 "
- »"''*"^ _. a; J__ r a;m + i
seeh-i (x/a) >
sech-i(a;/a) <
< X < a
-g < X <
1x1 > a
sinh~i
g m
14.673 j x"* cosh-* - dx =
14.674 I I-" tanh-' ^ dx =
•^ g
.4.675 J
m + 1
— - cosh-i
m + 1 a
^I^V^^^^
m
hi
a2
X>n + l
m
x"* coth-i — dx
a
14.676 r ajwi sech-' - dx =
-^ » ^:;^sech-^-^r^
14.677 C x'"csch-i^dx = -Ell^csch-i^ ± — ^^ r_£
•^ a m+1 am+ljr-
+ 1 ^ a^ ~ x^
dx
cosh-i (x/a) >
cosh*-i (x/a) <
m + 1
m+1 a m + 1 J ^^
X
+
7n
Vo^ - x^
'dx
\/g2 - x2
'dx
\/x2 + a2
sech - 1 (x/a) >
sech-'(x/o) <
[+ if X > 0, - if X < 0]
DEFINITE INTEGRALS
DEFINITION OF A DEFINITE INTEGRAL
Let f(x) be defined in an interval a ^ x S b. Divide the interval into n equal parts of length Aa; =
(b — a)/n. Then the definite integral of /(*) between x — a and x = b ia defined as
15.1 f fix)dx = lim {f{a)Ax + f{a + ^x)^x + /(a 4- 2Aa;) Ax + ■•■ + /(a + (n - 1) Aa;) Aaj}
The limit will certainly exist if f{x) is piecewise continuous.
If fix) = -raix), then by the fundamental theorem of the integral calculus the above definite integral
ax
can be evaluated by using the result
15.2 J f{x)dx = j i^s{.x)dx =
g{x)
= 9{h) - g{a)
If the interval is infinite or if fix) has a singularity at some point in the interval, the definite integral
is called an improper integral and can be defined by using appropriate limiting procedures. For example,
15.3 C f{x)dx = lim f fix)dx
15.4 f fix)dx = lim r f{x)dx
Xb y*b-e
fix) dx = lim 1 fix) dx if 6 is a singular point
15.6 I fix) dx = lim I fix) dx if a is a singular point
OENf^UFBRMfftA^NVofvTN^WBmTinwTfl
15.7
15.8
15.9
^b pi} fh fb
J (fix) ± gix) ± k{x) ± • • •> da: = J fix) dx ± J gix) da; ± J h{x) dx ± ■ ■ •
I c fix) da; = c j f{x) dx where c is any constant
f /(a;)da; =
15.10 r /(a;)da: = - C fix)dx
15.n f fix)dx = rf{x)dx+ C fix)dx
a a c
f(x) dx = (b — a) fie) where c is between a and 6
This is called the meayi value theorem for definite integrals and is valid if f{x) is continuous in
a^ x^b.
94
DEFINITE INTEGRALS
96
15.13
J /(x) g(x) dx ^ /(c) J g{x) dx where c is between a and 6
This is a generalization of 15.12 and is valid if f{x) and g{x) are continuous in a -^ x ^ h
and p(a;) ^ 0.
15.14
lEIBNITZ'S RUIE FOR DIFFERENTIATION OF INTEGRALS
*'*,ttt> »'*i(a) "" «« aa
^PSOXlWATrft^RMlilAlSoR DEFINITE^ »^^^
In the following the interval from x = a to x = b is subdivided into n equal parts by the points a = x^,
«!. X2, . . ., a:„_i, a:„ = 6 and we let y^ =^ fix^), y^ = /(a;,), y^ = /(arg), . . ., y„ = /(ic„), /t = (6 - a)/n.
Rectangular formula
15.
r''
IS J /(x) da: « A(yo + yi + y2+ •■■ + y„-i)
Trapezoidal formula
15.16
j;
fix) dx
^iVsi + 2j/, + 2i/2 + • • • + 2|/„_i + ?/„)
Simpson's formula (or parabolic formula) for n even
15.17
J f{x)dx « -g-d/o + 43/1 + 23/2 + 4^3 +-■■ +2i/„_2 + 4y„_i + !/„)
IN6 RATIONAL OR IRRATIONAL EXPRCSSIOI
15.18
15.19
15.20
15.21
15.22
15.23
15.24
15.25
X"
dx
x2 + a2
2a
X
sinpiT-
J„ 1 +
J'" x'^dx _
„ a;" + a" " n si
< p < 1
sin [(m + IV/n] '
< m+1 < n
x"^dx
jr sinmff
+ 2ar cos (i + x^ sin m-rr sin fi
dx
»^o y/cfi - x^ 2
Va2 - a;2 d-c -
I 3;"'(a" - a:")Pda: = j ^*' /
I (^"
x'^dx
vV[{m+\)/n-\- p + 1]
(-l)r-I^ttm + l-nrr[(m + l)/n]
+ a")'- nsin[(m + l)Wn](r-l)!r[(7n+l)/m - r + 1]
< mi + 1 < Mr
96
DEFINITE INTEGRALS
DEFINITE INTEGRALS INVOLVING TRIGONOMETR
All letters are considered positive unless otherwise indicated.
m,n integers and m ^^ n
m
S ''"
15.26 I sin Tnx sinnx dx =
1 jr/2 m, n integers and m = n
mi, M integers and m ¥= n
15.27 I cos mx cos nx dx == -s
7r/2 m, n integers and m = n
f
15.28 I ainmxcoanxdx —
m,n integers and m + n odd
2m/(m'^ ~ n^) m, n integers and m + n even
1 5.29 I sin2 xdx = i cos^ x dx ^ ^
sinS"" x dx
15.31 r sm2^+^xdx = P cosZ'^+ixdx = ^„'^_'^": ^^ . , m = l,2,...
''o
sin^P-ixcos^-i-ixdx = J^P^ ^<'^^
2r(p+(/)
15.33 r ?mp^d« = ^
cos2"'a:da: = ^ '■^ '.^ '." ^'!!~ ^ ^ . m = 1, 2, . . .
2 • 4 • 6 ■ • ■ 2mi 2
1 • 3 • 5 ■ ■ ■ 2m + 1
15.32
B-/2 p >
p =
-5r/2 p <
15.34
/:
sm px cos qx
x
dx —
I
15.35 f sin pa: sin ga;^^ ^
a^J
p > g >
:r/2 < p < qf
jr/4 p = g >
7rp/2 < p ^ g
7rg/2 p ^ Q >
15
sin^ px ^^ _ £p
- cos px .^ _ ffp
x^
da; = ^
.36 r°
15.37 ri
15.38 r"cospx-cosqa; ^ = i„ £
J, « p
15.39 r" cos px - cos qx ^^ _ Trig - p)
V
15.40 CSB^^dx = ^e--
J„ x2 + „2 2a
15.41
X°° X sinm
x2 + a:
mx ,„ _ 7r„_
dx - -^e-"*"
X x(x2 + a2)
- ^d-^-""")
15.43
15.44
15.45
dx
+ 6 sin X
n a
dx
'o a + 6 cos X
.ff/2
n a
dx
2n-
Va2-
62
2ff
Vo2-
62
COS"
(6/o)
+ 6 cos X yo2 — 62
DEFINITE INTEGRALS 97
15.46 r ' ^^ = C Ja; ^ 2r-a
J^ (a + 6 sin x)^ J^ {a + b cos a;)2 (a^ - 62)3/2
15.47 T" ^ ^ 2:r 0<a<l
15.48 f g sin a; da; ^ \ (v/a) \n (1 + a) \a\ < 1
J„ l-2acosx + a^ | . In (1 + 1/a) |a| > 1
,5.49 r cos.»xd. ^ _^a™^ ^,^ m- 0.1,2..
^Q 1 — 2a cos a; + a2 1 — a- - , ■
15.50 J sinoK^da; - J cos 0^2 da; = \'\[¥'
15.51 I sin aa:" da: = _1^ r(l/M) sin -^ , n > 1
15.52 I cosaa:" dx = — Vr l '1/") cos — , n > 1
J^ »mi/" 2t[
15.53 r^inx^^ ^ fsSlSdx = .fl
15.54 r^inx^ ^ „ , ^ :" _, 0<p<l
J(, a:P 2r{p) sin {pn-/2) '^
15.55 r^^^dar = ——J!——-, 0<p<l
Jg x» 2r(p) cos (pW2) ^
1 5.56 I sin ax^ cos 26a; da; — — ■\l-^ ( cos sin —
J(, 2 \ 2a \^ o a
15.57 I cosaa!^ cos26a: dx = 7: A /tt" I cos (- sin —
J^ 2 Y 2ay a a
"■58 J^ _^d. = -
15.59 r^il^dx = ^
Jo ^' 3
15.60 r^Hl^rf^ = i
J,. X 2
15.61
. 1"
dx
.Q . + tan"'x 4
,5.63 /;^H^.. = J,-i+l-i+..
^dx - I In 2
15.64 C ^iill^
15.65 p l-cosx ^^ _ rcosx^^ ^
Jo a; J^ X
1 5.66 I ( T — I — ■= — cos X 1 — = y
15.67 f^tan-ipx-tan-igx ^^ = £ ]„ £
J„ X 29
98
DEFINITE INTEGRALS
DEFINITE INTEGRALS INVOLVING EXPONENTIAL FUNCTIONS
15.68
15.69
15.70
15.71
15.72
15.73
15.74
15.75
15.76
15.77
15.78
15.79
15.80
15.81
15.82
15.83
15.84
15.85
15.86
Jg-ar cOs6x dz =
e-" sin bx dx =
X
X
e-"^' COS 6a: da; = i../?e-b«
2 \ a
a2+ (,2
&
a2 + 62
- e-°-sin6x ^^ = tan-i^
X a
dx = In -
-b'/4a
r"e-""' + f^ + -'da: = i - f^efb^-iacJ/to ^^ _6_
where erfc (p) = — | e-^ dx
C x'^e-'^dx = ^<^+l>
X'
*-4ac}/4a
_^ , _ r[(7n + i)/2]
x^e""* dx =
l'^-'"'*"^'^ =\y[l
gatm + i'/z
/■
xdx _ X-L-L-l-L + JLj-
e' - 1 12 "^ 22 "^ 32 "^ 42 "^
6
For even n this can be summed in terms of Bernoulli numbers [see pages 108-109 and 114-115].
r
xdx ^ 1 I . J^ _ A .
e^ + 1 12 22 32 42
12
For some positive integer values of n the series can be summed [see pages 108-109 and 114-115].
r
sinmx ^^ ^ 1 oth^- J-
g27rl _ X
2 2»i
DEFINITE INTEGRALS
15.87
15.88
/:
r^
X secpx 2 \a^ + p^
dx = tan~i tan~i -
CSC px p V
15.89 r"^-°^a-co5a^)rf^ ^ cot-la- fin (a^+l)
ITE INTEGRALS INVOLVING LOGARITHMIC FUNCTIOI
15.90 { x^(\nx)^dx = , ^ ,^^"'!' w> -1, n^0,l,2,...
If 717^0,1,2,... replace m! by r(n+l).
15.91 17
da: - ^2
+ X
15.92
x 6
J, 1-:
15.93 C'JLiLt^dx = 4
J, 0. 12
15.94 I'lnO^
dx = --
15.95 C lnx\n(l + x)dx = 2-21n2-^
15.96 r lnxln(l-
.-'ft
x)dz = 2 - J
15.97
15.98
15.99
pxP_nnx^^ = -irZcBCpjrCOtpTT 0<p<l
— dx — In —r
In X n + 1
I e-'\nx
dx = —y
V^
15.100 r e-^Mnxdx = -■^{7 + 2 1n2)
dx = -T-
4
dx = -5 In 2
15.102 I In sina; dx = I In cosx
(ln8inx)2dx = I (lneosx)2dir = |(ln2)2+|^
15.104 r xlnsinxdx = "T^^^
15.105
XTT/2
sin a; In si
sin X dx ~ In 2 — 1
J»27T /^2Tr
In (a + 6 sin x) dx = I In (a + 6 cos x) dx = 2o- In (a + V^^^)
(1 •'n
100
DEFINITE INTEGRALS
15.107 yin(a+bco^x)dx = ^ln(^£±v5!E^
15.108 j In (o2 - 2a6 C9S a: + 62) da: =
2n- In a, o ^ 6 >
2n- In 6, 6 ^ a >
15.109
15.110
15.111
I In (1 + tan x) dx - ^ In 2
Jf, 8
r sec a; In ( f^-^-^^^ ) rfa: ^ A{(cos-i a)2 - (cos-^ 6)8}
2.i„|,,. = _,i||i+?i||£H.EnSa^
See also 15.102.
DEFINITE INTEGRALS INVOLVING HYPERBOLIC FUNCTIONS
15.112
4^1;^ dx = ;^tanh^
Sinn ox 26 26
15.113
X
15.114 f
COS ax , _ TT . oir
— ; — ; — «* = :rr seen rrr-
cosh hx 26 26
sinh aa: 4a2
15.115
fii
a;" da; _ 2"+i - 1 . . .,. | 1 . 1 1
sinhaa; ~ 2"a«+i ^ I" "^ -^M x-i + i "^ 2"+' "^ 3» + » "*
If M is an odd positive integer, the series can be summed [see page 108].
15.116 r^^dx = ^C8c^-f
I e^x -1-1 26 6 2a
15.117 r"^h«^da: = f-^cot¥
J ebz _ 1 2a 26 6
ISCELLANEOU5 DEFINITE INTEGRAL
M
,5., ,8 J^/W^AMrf^ ^ (/(O) -/{-)} In ^
This is called Frullani's integral. It holds if fix) is continuous and I '^^' ~ '^'^' dx converges.
15.120 r (a + a;)'"-i{a-a:)''-ida; = (2a)'" + n-ilMi:M
^~„ rim + n)
16.1
16.2
16.3
DEFINITION OF THE GAMMA FUNCTION T{n) FOR n>0
r(
n) = ( f-^e-
^ n
*dt « >
r(n + l) = nr{n)
r{n + l) = nl if n - 0,1,2, ... where 0! - 1
THE GAMMA FUNCTION FOR n<0
For n < the gamma function can be defined by using 16.2, i.e.
16.4
n
GRAPH OF THE GAMMA FUNCTION
r(M)
_ n .' 1
U-^ t
^^t -/
— 1 -S -1 -1 1 > S 4 I
„ -I
it ^ _.
Fig. 16-1
16.5
16.6
16.7
SPECIAL VALUES FOR THE GAMMA FUNCTION
rim+i) = l'3>5.^-^(2n.-l) ^ ^. 1.2.3,
(_l)m2m^
r(-» + i) = ^^3^^
(2m - 1)
m- 1,2,3,
101
102
THE GAMMA FUNCTION
RELATIONSHIPS AMONG GAMMA FUNCTION!
16.8
16.9
r(p)r(i-p) =
sin pn-
22*-* r{x) r{x + ^) = V^r(2x)
This is called the duplication formula.
16.10
For m = 2 this reduces to 16.9.
OtHErDteFrMlttONS OF THE GAMMA PUNCtrON
16.11
16.12
^^^+^^ = ji"L (.+iK;V2V---t+fe)
fe*
This is an infinite product representation for the ^amma function where y is Euler's constant.
DERIVATIVES OF THE GAMMA FUNCTION
16.13
:"(!) = j e-* In K da: = -y
16.14
r(x) ^ ll =»/ \2 *+ 1
+ ■■- + ^-
n X + n~ 1
+
16.15
^NS FOR THE Gi
r(« + l) = ■\/2vXX='e-^ ^1 + r^ + ^
139
12x 288a;2 51,840x3
+
This is called Stirling's asymptotic series.
If we let X = n a positive integer in 16.15, then a useful approximation for n\ where n is large
[e.g. 71 > 10] is given by Stirling's formula
16.16 n\ ~ \/2^n«e-''
where — is used to indicate that the ratio of the terms on each side approaches 1 as n-* «>.
16.17
|r(tx)|2 =
X sinh vx
DEFINITION OF THE BETA FUNCTION B{m,n)
17.1
B(m,n) = J (^-^l-t)"-i dt m > 0, n >
RELATIONSHIP OF BETA FUNCTION TO GAMMA FUNCTION
17.2 B{'m,n) = iMliZii
r(m + n)
Extensions of B(m, n) to m < 0, n < is provided by using 16.4, page 101.
ME IMPORTANT RESULTS!
17.3
B{m, n) = B{n, m)
17.4
B(m
,n) = 2 I
ir/2
gJn2n"-l « cos^"-! 9 rffl
17.5
B(m
'"^ - X (l+V-^«
d«
17.6
103
18
BASIC DIFFERENTIAL EQUATIONS
and SOLUTIONS
...^Mmmi^^dLMOisAnou
SOIUT.ON : , :!Hi^
18.1 Separation of variables
1 , , .dx + 1 —r^dy - c
fi(x)giiy)dx + f%(x)g2{y)dy =
18.2 Linear first order equation
ye/Pdx = j Qe Jpditia; + c
1^ + P(x)y - Q(x)
18.3 Bernoulli's equation ■- ;'
where i" = y^""". If n = \, the solution is
In y = \ {Q-P)dx + c
g + P(x)y = Q{x)y^
18.4 Exact equation
\ Mdx + (In - J- ( Mdx\dy = c
where Bx indicates that the integration is to be performed
with respect to x keeping y constant.
M{x, y) dx + N{x, y)dy =
where dM/dy = dN/dx.
18.5 Homogeneous equation
, C dv ^
J F{v) - V
where v = y/x. If F{v) = v, the solution is y =^ ex.
f = <f)
104
BASIC DIFFERENTIAL EQUATIONS AND SOLUTIONS
105
OlFFiRENTIAL EQUATION
18.6
y F{xy) dx + X G{xy) dy —
!• ^ Linear, homogeneous
second order equation
d^y , dy
dx^ dx
a, b are real constants.
^ + a— + 61, =
18 8 Linear, nonhomogeneous
second order equation
a, 6 are real constants.
SOLUTION
■'(
In a: - f ^-^M^1L_ + „
'""^ " J v{Giv)-Fm^'
where v = xy. If G(v) = F(v), the solution is xy = c.
Let mi,m2 be the roots of m^ + am + b = 0. Then
there are 3 cases.
Case 1. m,, m^ real and distinct:
y = Cie*"i* + cge""!*
Caae 2. m,, mg real and equal: " ' '
y = c,e"»i* + C2a:e"'i^
Case 3. rtii = p + gi, m2 — p — qv.
y = eP*(ci cos qx + Cj sin qx)
where p = — o/2, g = \/6 — oV4 .
There are 3 cases corresponding to those of entry 18.7
above.
Case 1.
y = Cie*"!* + Cg^"**
mi — 7112 J
WI2 ~ '"i ^
Case 2.
+ see"!* I e~'"i^fi{x)da:
— e*"!* I xe-'^i'^ R(x) dx
Cases.
y = ei*E(cj COS qx + e^ sin qx)
+ ^— I e P- «(a:) cos oa; da;
18.9 Euler or Cauchy equation
S + -f + *. = S(.)
Putting X = e', the equation becomes
and can then be solved as in entries 18.7 and 18.8 above.
106
BASIC DIFFERENTIAL EQUATIONS AND SOLUTIONS
IH^DIFFERENTIAL EQUATION
SOLUTION
18.10 Bessel's equation
y = CiJ„(Xa) + C2y„(a!)
See pages 136-137.
x^^ + x^+ {\^x--^-n^)y =
18.11 Transformed Bessel's equation
y = x-r>j^e,J^,J^xA + cY^J^x^Xi
x2 + (2p + l)a^ ^ + {a^x^^ + P^)y =
where q = Vp^ ~ P^ •
18.12 Legendre'a equation
y = eiP„{x) + c^Q^lx)
See pages 146-148.
^^-''^^2-2x^ + ^i^ + i)y =
IF-
i^ '
RITHMETIC SERIE!
19.1 a + ia + d) + {a + 2di + ■■■ + {a+ in-l)d} = ^n{2a + (n - l)d} ^ ^n{a-i-l)
where 1 = a+{n — l)d is the last term.
Some special cases are
19.2
19.3
l + 2 + 3+---+n = |n(n + 1)
1 + 3 + 6 + •■• + {2w-l) = m2
19.4
EOMETRIC SERIES^
a + ar + wfi + a-fi + • ■ • + ar"""^
where ? = ar«-' is the last term and r t^ 1.
a(l - r")
1 - r
g — rf
1 -r
If -1 < r < 1, then
19.5
a + ar + ar^ + or^ +
1 -r
19.6 a + (a + d)r + (a + 2d)r2 + •-■ + {a+(n-l)d}rn-i = a(l-r«) ^ rd{l - nr"-i + (n- l)r"}
1 - r (1 - r)2
where r ¥^\.
If -1 < r < 1, then
19-7 a + (o + rf)r + (a + 2d)r2 +
r(f
1 - r ' (1 - r)2
SUMS OF POWERS OF POSITIVE INTEGERS
19.8 IP + 2" + 3P + •■■ + „p = ^^ + inp + ^^P" _
P + 1 2 2!
BipW-i B2P(P - 1)(P - 2)nP-3
4!
+
where the series terminates at n^ or n according as p is odd or even, and B^ are the Bemoullx
numbers [see page 114].
107
108
SERIES OF CONSTANTS
Some special cases are
19.9 1+2 + 3+ ■•■+n = "tn+D
2
19.10 12 + 22 + 32 + ... + n2 = Mn + l)i2n + 1>
6
19.11 13 + 23 + 33 + ■•■ + n3 = nHn + l)i ^ (i + 2 + 3 + • ■ • +n)2
19.12 14 + 24 + 34 + ... + „4 = «(n + l)(2n + 1)(3«2 + 3n - 1)
30
If Sfc — l** + 2"= + Sif + • • • + n'^ where k and n are positive integers, then
'^ + l\« . . /k+l
k
k + \
1
19.13 C: ^ iSi + [■■;" )Sa +•••+[-;* )Sk = (n+ 1)^+1 - (n+l)
19.14
19.15
19.16
19.17
19.18
19.19
19.20
19.21
19.22
19.23
19.24
19.25
19.26
19.27
19.28
19.29
19.30
19.31
19.32
2^3 4^5
3^5 7^9
4^7 10 ^ 13
5^9 13 ^ 17
= ln2
^+i
11 14
+
1
44
+
+
1
48
+
-
1
42
+
-
1
4*
+
-
1
46
+
+
1
72
+
+
1
74
+
l_l+i_
2 5 8
J- + J-+J_
12 ^ 22 32
14 ^ 2^ 3«
1+1 + X
16 ^ 2« 3«
1-1 + 1
12 22 32
1-1 + 1
14 2* 3*
1_1 + 1
1« 28 38
1 + 1 + 1
12 ^ 32 52
1 + 1 + 1
1* 3* 5^
1+1+1 + 1 +
16 ^ ga ^ 5« 76 ^
1_1+1_1+
13 33 53 73
1 + 1_1_1 +
13 ^ 33 53 73 ^
1*3 ^ 3-5 ^ 5-7 ^ 7-9
^ + ^ + -^+^ +
1'3 2-4 ^ 3-5 ^ 4-6
rV| , V2 In <1 + V^)
8 "^ 4
6
90
_£l.
945
i^
12
7^
720
^ 31irB
30,240
8
96
960
zi
32
3ir2\/2
• ■•-■^C J' -'. .'■•■
1 + 1,+ 1,+ 1,+
16 •• -.7 .
_ 1
2
3
4
12.32 ' 32.52 ' 52*72 ' 72*92
16
SERIES OF CONSTANTS
109
19.33
+
+
12,22.32 22 -32 -42 32-42.52
1 1
+
47^2 - 39
1».34 i 1 ^ ,
a a + d a + 2d a + 3d
+
16
1 ..«-
+
19.35 J-+J^+J^+J_ +
12p ^ 221' 32P ^ 42p ^
19.36 J-+J_+JL+J_ +
12P ^ 32p ^ 52p ^ 72p ^
19.37 i--J_ + J-_J_ +
l2p 22P S^K 42P
X l + «
22P-l^PBp
(2p)!
{22P - l),r2PBp
2(2^)]
(22p-i-i)^PBp
du
d
{2p)\
19.38
^ + 1
12P+1 32P+I '^ 52P+I 72P+I
+
;r2p + lSp
22p + 2(2p)J
n
r'^A
MISCELLANEOUS SERIES
19.39 l + COSa + C0S2«+ ■-. + COSTla :3: ^^" ^^ ^ ^^^
2 2 sin (a/2)
19.40 sin« + 3in2a + 8m3a+ ••• +sinn« = sin [^(n + !)]„ sin^n«
sin (a/2)
19.41 1 + rcosa + r2cos2a + r3 cos 3a +
1 — r cos g
X — 2r cos a + 9-2 '
|r| < 1
'• .H
19.42 rsina + r2am2a + r«sin3a + ■•• = r_^Tia ,, .
1 -2rcosa +r2' I'^l ^ ■"
19.43 1 + rcosa + r2cos2a + ••■ + r«coan« = r'*'^^ cos na - r" + i co3(n + l)a - r coaa + 1
1 - 2r cos a + r2
19.44 rsina + 7^sin2a + ■•■ + r^sinTta = ^si""-^""^^ sin(n + l)a + r"+2 8inna
1 — 2r cos a + r^
hit
THE EUIER-MACLAURIN SUMMATION FORMULA
19.45
n-l
fc
2 F{k) = f F(k) dk ~ I {F(0) + F(n)}
+ ^{F'(w)-F'(0)} - tIo^^'"**'*"^'"^*'^
A. i
+
+ ... (-l)p-l_^{i^(2p-I)(„) _p(2p-l)(0)) + ...
19.46
THE POISSON SUMMATION FORMULA
2 F(fc) = 2 11 «27Kn«/r(a.) d:c
{/:
if
20.1
TAYiOR SERIES FOR FUNCTIONS OF ONE VARIABLE
2! (n— 1)! "
where R„, the remainder after « terms, is given by either of the following forms:
20.2 Lagrange's form ie„ = /^"H$)(a; - a)"
20.3 Cauchy's form
_ /t">a)(x~{)"-Hx-a)
The value $, which may be different in the two forma, lies between o and z. The result holds if f(x) has
continuous derivatives of order n at least.
If lim fi„ = 0, the infinite series obtained is called the Taylor series for f(x) about x = a. If
a = the series is often called a Maclaurin aeries. These series, often called power series, generally
converge for all values of x in some interval called the interval of convergence and diverge for all x outside
this interval.
BINOMIAL SERIES
20.4
(a+«)" = g" + na^-^x + "t"^~ ^> an-2a.2 + ^(" " IK"" 2) an-3a;3 +
= a" +
Special cases are
a^-ix +
a«-2a:2 +
20.5 (a+a;)2 = a^ + Zaz + x^
20.6 {o + a;)3 - a^ + Sa^x + Saz^ + x^
20.7 (a + 3:)^ = a4 + 4a3a: + 6a2x2 + 4aa;3 + ar*
20.8 (l + a:)-i = 1-x + x^-x^ + x*----
20.9 (l + a)-2 = 1 — 2a; + 3a;2 - 4x3 + 5a^ - • ••
20.10 (l + a!)-3 ^ 1 - 3x + 6x2 - 10x3 + 153^ _ ..
20.11 (l+x)-/2 = i_l^+|l|,2- 113^^3 +
20.12 (H-x)»/2 = i+±x-^x^ +
1-3
2'4'6
l-4'7
20.13 (1 + x)— . i_l, + ll|,. ___
X3 - -
X3 +
20.14 (l + x)./3 = 1-,!, --!_,. + _il^,3_
(in-3a;3 +
-1 < X < 1
-1 < X < 1
-1 < X < 1
-1 < X ^ 1
-1 < X ^ 1
-1 < X ^ 1
-1 < X ^ 1
110
TAYLOR SERIES
111
20,15 e' = i + x+|J + f^+ ...
= ^n + f + T +
-1< X < 1
20.19 Inx = 2U^^] + ^^^ +lf?^^ll
x+l/ '^ 3[x + lj "'■fltTf' "*"
— « < X < *
20.16 a- = e-in« = 1 + « ]„ a + i^il^ + t^J^L^ + . . . -« < ar < «
20.17 ln(l + x) = a;-^ + ^-^+... -Kxfil
23 4 i%x = j.
20.20 ,„. = f^)+|(E^y,|(^l-
+
X >
x^i
;eries for trigonometric functions
20.21 sinx
20.22 cos X
20.23 tan z ■■
20.24 cot X --
20.25 sec x
20.26 CSC X =
20.27 sin- Ik
20.28 cos~ix
X^ . X^
2! ^4! 6!
1 _ X _ x^ 2x°
X 3 45 945
"^ 2 "^ 24 ■*" 720 "^
X 6 360 15,120 "•■
— 00 < a; < oo
— ~ < X < «=
+ 1 —2 +
(2n)!
22nB„K2n-l
+
(2n)!
+
2(22''-'-l)B„x2''-'
(2n)!
= a: +i^+il3xf 1^3^x7
2 3^ 2"4 5 ^ 2-4*6 7
— jr ._, »■ / ,1x3 1-3x5,
20.29 ten-»x =
r 3+5 7 +
U--i + J L +
^2 X 3x3 5a.5 ^
w<|
< |xl< »•
w<t
20.30 cot-'x = ^-tan-ix =
x^ . x=
2-(^^-T + T- ■
P" + x-3^+5^-
20.31 sec-i X
20.32 csc-ix
2 V X 2 • 3x3 ^ 2 • 4 • 5x6
+
sin-> (1/x) = - + ^— + ^'^
X 2 • 3x3 ^ 2 • 4 • 5x5
+
< |xl < 0-
|x| < 1
\x\ < 1
|x| < 1
[+ if X £ 1, - if X ^ -1]
lx| < 1
[p - if X > 1, p - 1 if X < -1]
|xt > 1
Ixl >1
112
TAYLOR SERIES
SERIES FOR HYPERBOLIC FUNCTIO
X^ , K"
2t ^ dt ^ fit ^
4!
^"T + ir
6!
315
+
ic 3 45 945
x2 . 5^4 61k6
1 - — +
2 ^ 24
+
± f In 12x1 +
20.39 sinh-ia; =
20.40 cosh-' X = ± -( In (2x) -
20.41 tanh->x
20.42 coth-»iB
720
31x5
15,120
„ _ JEL + 1-3x5
2-3 ^ 2-4.5
1 X 7x^
X 6 "^ 360
+
— '^ < X < *
— OS < X < «
(_l)n-122n(22n_l)B^a;2t>-I
(2^0!
(-l)''-122«B„x2n-l
(-1)"£„X2"
■ (2«)I ■*" ■■■
(-l)«2(22n-i - l)5„x2n-i
+
1 • 3 • 5x'
2*4'6-7
1'3
+
2*2x2
1
+
(2n)!
1'3'5
+
2 • 4 • 4x4 ^ 2 • 4 • 6 • 6x8
+
1*3 I l'3-6
2-2x2 2-4'4x* 2-4-6-6xfl
^^f^y^y^
a; 3x3 ^ 5a.5 ^ 73.7
■)} [:;
1.1 <i
< 1x1 <
w<i
< |x| < a-
lx| < L
"+ if X ^ 1
- if X ^ -
if cosh-i X > 0,
if cosh-' X < 0,
|x|<l
\x\ > 1
.]
X ^
X
:a
MISCELLANEOUS SERIES
20.43 e»- = i+:c+f-f-^+...
20.44 e'^°^ *
, , x2 a:* 31x8
20.45 efn* = 1 +x + ^+ ^ + 3£l+ ...
^ A O
20.46 e-sinx = x + x2+%^-^-^+ ... + 2:!^iliniW4Ix^^ ...
a oU 90 7(,!
20.47 e-cosx = l + x- ^-^+ ■-- + 2"^^ cos (n./4) x» ^ ...
3 D
n!
on ^« 1 I • I lit ^^ a^ 3:8
20.48 Inlamxl = m |xl - - - — - ^835 "
20.49 inlcosxl = -^ - ^^ - g - ^^^^
22"-iB„x2«
w(2m) ! "^
2a«-i(22«-l)g„g2"
7i(2ti) !
+
on en 11* 1 1 1 1 . «^ . ^ar* , 62x«
20.50 ln|tanx| = in\x\ + ^ + -^ + ^^
+
22'>(22"-i-l)B„x2"
n(2n) !
20
.51 llLikL£) = jc - (l + ^)x2 + (l + ^ + i)x8 - -■-
X "i X
— <* < X < <«
— *o < X < <»
I r ^ "
W<2
— =° < X < •
— «° < X < »
< Ixl <
N<^
< kl< I
1x1 < 1
TAYLOR SERIES
113
If
20.52
then
20.53
where
20.54
20.55
20.56
20.57
20.58
20.59
y = Cix + c^x^ + Cgx^ + c^^ + CsxS + c^jje + . . .
a: = Ciw + Czv" + C^y^ + C^y* + €^y^ + Cei/S + • - •
c^C^ = 1
c\C^ — Bcic^e^ — 5<^ — eje4
e^Cs = 6<!?C2C4 + 3cfc2 _ cjcj + 14c^ - 21c,c|c3
cJ^Cfl = Tc^cgCs + 84c,c|c3 + Tcf^^^ _ 28c2c2c2 - c^^c^ - 2Sclelc^ - 42c|
TAYIOR SERIES FOR FUNCTIONS OF TWO VARIABLES
20.60 f(x,y) = f{a.b) + (x-a)fMb) + iy-b)f^{a.b)
+ -^(i^- aW^ia. b) + 2(x - a){y - b)f^(a, b) + {,y- 6)»/„(o, 6)} + • • •
where fjfl., b), fy{a, b), , . . denote partial derivatives with respect to ar, j/, . . . evaluated at x = a, y = b.
DEFINITION OF BERNOULLI NUMBERS
The Bernoulli numbers Bj, B^, B^, . . . are defined by the series
21.1 '^
e*- 1
= 1 _ ^ 4_ ^1*' ^2^* ^3^:8
2'^ 2\ "IT "^ ~6r
x\ < 27
21.2
, X ^x Bix^ B2X* BgajB
1 — « cot - = — : 1- — 1 = — +
4! ^ 6!
2!
I»l <^
DEFINITION OF EULER NUMBERS
The Elder numbers E^,E^E^, ... are defined by the series
21.3 s«.hx = 1-^ + ^.^ +
2!
4!
6!
W<|
21.4
secx
= 1+^ + ^+^ +
21
4!
6!
W<2
BERNOULLI AND El
llfiRS
Bernoulli numbers
Euler numbers
Bi = 1/6
El
= 1
B2 = 1/30
E2
= 5
B3 = 1/42
Ez
= 61
B4 = 1/30
E,
= 1385
B5 = 5/66
E-,
~ 60,521
Bfl = 691/2730
Ee
= 2,702,766
B7 = 7/6
E,
= 199,360,981
Bg = 3617/510
En
= 19,391,512,145
B9 = 43,867/798
E^
= 2,404,879,675,441
Bio = 174,611/330
Eio
= 370,371,188,237,525
fill = 864,513/138
En
= 69,348,874,393,137,901
B12 = 236,364,091/2730
E12
= 15,514,534,163,557,086,905
114
BERNOULLI AND EULER NUMBERS
115
JmBwW i W oi» BtU'lffC'yi'tnwB^^HiHffluwWR
2n+ 1
2
2n+ 1
21.5 ( „ ")22Bi - ^^ 4 "J24B2 + [ g ~ )2«B« -
2n+ 1
(_l)n-i(2n + 1)2^B„ = 2n
21.6
2n f/2n - 1
*'•' ^» = ^.=(1^ |(^'"r V ^»-' - ('"3" ^"- + ('"5- ' ) ^"-3 - • • ■ (-«-
SERIES INVOLVING BERNOULLI AND EULER NUMBERS
21.8 B^
- (2n)!
l + i + a= +
22n-l^nl 22" 32«
21.9 B„ =
2(2?i) !
1+ 1 + 1 +
(22" - l)n^n ] 32" 52"
21.10 B, =
(2n)! __L+ J_
21.11 £;„ =
22n+2(2n)! K 1_ 1
y2n+l 1^ 32n + l "•" 52n + l
21.12
B. ~ in^Hirei-^'^Vrn
FORMULAS from
VECTOR ANALYSIS
VECTORS AND SCALARS
Various quantities in physics such as temperature, volume and speed can be specified by a real number.
Such quantities are called scalars.
Other quantities such as force, velocity and momentum require for their specification a direction as
well as magnitude. Such quantities are called vectors. A vector is represented by an arrow or directed
line segment indicating direction. The magnitude of the vector is determined by the length of the arrow,
using an appropriate unit.
NOTATION FOR VECTORS
'£
A vector is denoted by a bold faced letter such as A [Fig. 22-1]. The magnitude is denoted by |A|
A. The tail end of the arrow is called the initial point while the head is called the terminal point.
or
2.
3.
FUNDAMENTAL DEFINITIONS
1. Equality of vectors. Two vectors are equal if they have the same
magnitude and direction. Thus A = B in Fig. 22-1.
Multiplication of a vector by a scalar. If m is any real number
(scalar), then mA is a vector whose magnitude is \m\ times the
magnitude of A and whose direction is the same as or opposite
to A according as m > or m < 0. If in = 0, then mA = ia
called the zero or null vector.
B.
Fig. 22-1
Sums of vectors. The sum or resultant of A and B is a vector C = A + B formed by placing the
initial point of B on the terminal point of A and joining the initial point of A to the terminal point
of B [Pig. 22-2(6)]. This definition is equivalent to the parallelogram law for vector addition as in-
dicated in Fig. 22-2(c). The vector A-B is defined as A + {-B).
1X6
FORMULAS FROM VECTOR ANALYSIS
117
Extensions to sums of more than two vectors are immediate. Thus Fig. 22-3 shows how to obtain
the sum E of the vectors A, B, C and D.
(a)
D
Fig. 22-3
(b)
4. Unit vectors. A unit vector is a vector with unit magnitude. If A la a vector, then a unit vector in
the direction of A is a = A/A where A > 0.
LAWS OF VECTOI
tEBRA
If A, B, C are vectors and m, n are scalars, then
22.1 A + B = B + A Commutative law for addition
22.2 A + (B + C) = (A + B) + C
Associative law for addition
22.3 m(7iA) = {mn)A = n(m.A) Associative law for scalar multiplication
22.4 (m + 7i)A = mA + nA Distributive law
22.5 mi{A + B) =^ »iA + rniB Distributive law
COMPONENTS OF A VECTOR
A vector A can be represented with initial point at the
origin of a rectangular coordinate system. If i, j,k are unit
vectors in the directions of the positive x, y, z axes, then
22.6
A = Ax\ + AJ + A^
where Aii.AaJ.Agk are called ctym'poneni vectors of A in the
i, j, k directions and A^.A^, A3 are called the components of A.
DOT OR SCALAR PRODUCT
Fig. 22-4
22.7
A • B = AB cos * ^ S S
where e is the angle between A and B.
118
FORMULAS FROM VECTOR ANALYSIS
Fundamental results are
22.8 A*B = B-A
22.9 A-(B + C) = A-B + A-C
22.10 A-B = AiB^+ A2BZ + A3B3
where A = Aii + A^ + Agk, 3 = 811 + 823 + 8^^.
Commutative law
Distributive law
CROSS OR VECTOR PRODUCT
22.11
A X B = AB sin S u
0^ S Sir
where $ ia the angle between A and B and u is a
unit vector perpendicular to the plane of A and B
such that A, B, u form a right-handed system [i.e. a
right-threaded screw rotated through an angle less
than 180° from A to B will advance in the direction
of u as in Fig. 22-5].
Fundamental results are
22.12 AXB =
i j k
Ai Az A3
Bi Bj ^3
Fig. 22-5
22.13 AXB = -BXA
22.14 AX(B-I-C) = AXB-I-AXC
22.15 |A X B] = area of parallelogram having sides A and B
ISCELtANEOUS FORMULAS INVOLVING DOT AND CROSS PRODUCl
22.16 A'(BXC) =
Aj A2 A3
81 82 8^
Ci C2 C^
= AjfiaCa + Aa^aCi + As^iCg - AgBgCi - AjjBiCg - AjBaCj
22. 1 7 I A • (B X C) [ = volume of parallelepiped with sides A, B, C
22.18 AX(BXC) = B(A>C) - C(A'B)
22.19 (A X B) X C = B(A • C) - A(B • C)
22.20 (A X B) • (C X D) = (A • C)(B • D) - (A • D)(B • C)
22.21 (A X B) X (C X D) - C{A • (B X D)} - D{A • (B X C)>
= B{A • (C X D)} - A{B • (C X D))
_
FORMULAS FROM VECTOR ANALYSIS
119
DERIVATIVES OF VECTORS
The derivative of a vector function A(m) = A i(u)i + A2(u)j + A^iu^k of the scalar variable u is
given by
„-, dA ,. A(M + AM)-A(tt) (Ml dAa dA^
22.22 -J— = hm = -;— i 4- -;— j + -;— k
du iu-*o Au du du du
Partial derivatives of a vector function A(a;, y, z) are similarly defined. We assume that all derivatives
exist unless otherwise specified.
RMULAS INVOLVING OERIVATIV£$
22.23 #(A.B) = A.^+^-B
du du du
22.24 -^(AXB) = AX^ + ^XB
du du du
22.25 £iA.(BXC)> = ^.(BXC) + A. f XC+A.fBxf
22.26 A.^ = A^
du du
22.27 A ■ -r~ = if [A[ is a constant
The operator del is defined by
22.28
THE DEL OPERATOR
V = l~ H ]T h k-r-
dx dy dz
In the results below we assume that U = U{x, y, z), V = V(x, y, z), A = A(«, y, z) and B = B(a:, y, z)
have partial derivatives.
THE GRADIENT
22.29 Gradient of U - grad U -
VU = {\^ + \±.+ k^)U
'3a: * by dz '
dU . ^ dU. _L 3C7.
5a; dy dz
THE DIVERGENCE
22.30 Divergence of A = divA = V-A =
's+'^ + ''i)-'^'' + ^^ + -*^'"
SAi dAz dAs
dx dy dz
120
FORMULAS FROM VECTOR ANALYSIS
THE CURL
22.31 Curl of A = curl A = V X A
*l; + i^ + »'i)=^<^^*-^^^ + ^3k)
i j k
-1 A A
5a; dy dz
dy
dz
dA2\ fdAi SAgN fdAz dAi
THE LAPLACIAN
22.32 LaplacianofU = V^C/ = V'[VU) = ^+^ + ^
22.33 LaplacianofA = v^A = + + ^
THE BIHARMONIC OPERATE
22.34 Biharmonic operator on C7 = ^*U = V^V^U)
22.35 V(U + V) = VU + VV
22.36 V'(A+B) = V'A+V'B
22.37 VX(A + B) = VXA+VXB
22.38 V-(t/A) = (VC/)'A+ [/(V'A)
22.39 V X {UA) = iVU)XA + U(V X A)
22.40 V'(AXB) = B'(VXA) - A-(VXB)
22.41 Vx(AXB) = (B"V)A - B(V-A) - (A*V)B + A(V'B)
22.42 V(A-B) = (B*V)A + (A*V)B + BX(VX A) + AX(VXB)
22.43 V X (V IT) = 0, i.e. the curl of the gradient of t7 is zero.
22.44 V • (V X A) = 0, i.e. the divergence of the curl of A is zero.
22.45 V X (V X A) = V(V • A) - V2A
FORMULAS FROM VECTOR ANALYSIS
121
If A(m) = ^^^(w)' *^C" t*^6 indefinite integral of A{u) is
22.46 I A(m) du = B(it) + c c ::= constant vector
The definite integral of A(w) from u = a to u = & in this case is given by
22.47 r Aiu)du = B(6) - B(a)
The definite integral can be defined as on page 94.
IE INTI
Consider a space curve C joining two points Pi{ai,a2,a^) and
^2(^1. &2. ''3) as in Fig. 22-6. Divide the curve into n parts by points
of subdivision {xi.y^.Zi), .. .,(a:„_i,y„_],z^_i). Then the line integral
of a vector A(a:, y, z) along C is defined as
22.48 f A-dr = r 'A-dr = lim 1 \{x„,y^,z^)' ^r^
where Atp - AKpi + A?/p j + Az^k, AXp = a!p+ 1 - aip, Ai/p = i/p + j - ^p,
A«p = Zp+i — Zp and where it is assumed that as n -» «= the largest
of the magnitudes |Arp| approaches zero. The result 22.48 is a gen-
eralization of the ordinary definite integral [page 94].
The line integral 22.48 can also be written
22.49 I A-dr = \ Aydx + A^dy + A^dz
using A = All + Agj + Agk and dt — dxi + dyj + dzk.
Fig. 22-6
PROPERTIES OF IINE INTEGRALS
22.50
22.51
A • dr = - J A • dr
'-'1 P|
rPi r^s /-pj
I A • dr = ) A • dr + I A • dr
-'p, *^Pj "^Pa
NDEPENDENCE OF THE PATH
In general a line integral has a value which depends on the particular path C joining points Pj and Fj
in a region %. However, in case A = V^ or V X A = where <f> and its partial derivatives are con-
tinuous in %, the line integral I A • dr is independent of the path. In such case
22.52 r A • dr = f ' A • dr ^ 0(^2) - ^(^1)
122
FORMULAS FROM VECTOR ANALYSIS
where ^(Pi) and 0(^2) denote the values of at Pi and Pa respectively. In particular if C is a closed curve,
X
£
22.53 I A'dr = * A-rfr ^
c *^r
where the circle on the integral sign is used to emphasize that C is closed.
MULTIPLE INTEGRALS
Let F(x, y) be a function defined in a region % of the
xy plane as in Fig. 22-7. Subdivide the region into n parts
by lines parallel to the z and y axes as indicated. Let AAp =
Aajp i|/p denote an area of one of these parts. Then the in-
tegral of Fix, y) over % is defined as
22.54 r F{x,y)dA =^ lim "2 F(Xp,yj,) AA,
J io n-*« P=l
provided this limit exists.
In such case the integral can also be written as
I F(«, y) dy dx
i F{x,y)dy\dx
x=a i'^v=f,(x) J
where y = fi(x) and y - fzix) are the equations of curves PHQ and PGQ respectively and a and b are
the X coordinates of points P and Q. The result can also be written as
d
y
Aaip i!/p
.A
^
G
r-^
n
s.
yp+i
<
s
•v
^.
\
J
%
\q
K
e
1
1
^
H
\
a
X
p X
p+i
h
Fig. 22-7
22.56
I f(a:,ff)dicdi/ ^ I 1 I FKX,y)dx\dy
« = C ^1 = 01(1*) •-'u = C L''i = gi(!() J
where x = g^iy), x = ffziy) are the equations of curves HPG and HQG respectively and c and d are the y
coordinates of H and G.
These are called double integrals or area integrals. The ideas can be similarly extended to triple or
volume integrals or to higher ■multiple integrals.
SURFACE INTEGRALS
Subdivide the surface S [see Fig. 22-8] into n elements of
areaiSp, p=l,2,...,n. Let A(a;p, yp, Zp) = Ap where (Xp, j/p, 2p)
Let Np be a unit normal to ASp at P.
is a point P in iSp.
the surface integral of the normal component of A
defined as
Then
over S is
22.57
X
A-tidS =
lim 2 Ap
n-*iB p=l
N„AS„
Fig. 22-8
FORMULAS FROM VECTOR ANALYSIS
128
RELATION BETWEEN SURFACE AND DOUBLE INTEGRALS
If ^ is the projection of S on the xy plane, then [see Fig. 22-8]
22.58
Jo J J N-k
%
THE DIVERGENCE THEOREM
Let £r be a closed surface bounding a region of volume V; then if N is the positive (outward drawn)
normal and dS = N dS, we have [see Fig. 22-9]
22.59 I VA dV = I A'dS
The result is also called Gauss' theorem or Green's theorem.
f V • A dF = r
•-'it ^ e
J
Fig. 22-9
Fig. 22-10
STOKE'S THEOREM
Let S be an open two-sided surface bounded by a closed non-intersecting curve C [simple closed curve]
as in Fig. 22-10. Then
22.60 4) A • dr = I {V X A) • dS
where the circle on the integral is used to emphasize that C is closed.
£ A'dr = r {V X A)
GREEN'S THEOREM IN THE PLANE
22.61
£ Pdx + Qdy = C [^ -^\ dxdy
J^ Jj, \3x dyj
where R is the area bounded by the closed curve C. This result is a special case of the divergence theorem
or Stoke's theorem.
124
FORMULAS FROM VECTOR ANALYSIS
22.62
GREEN'S FIRST IDENTITY
J {^W + (v^)-(v^)>dv = ^{^v^)
dS
where p and ^ are scalar functions.
GREEN'S SECOND IDENTITY
22.63
f (0VV - <pV^)dV = f (0V^ - vV0)-dS
22.64
X
V X A dV =
= 1
(fSX A
22.65 f ^dr = J dSXV0
CURVILINSA,R COORDINATES
A point P in space [see Fig. 22-11] can be lo-
cated by rectangular coordinates {x, y, z) or curvi-
linear coordinates (Mj, M2» "s) where the transforma-
tion equations from one set of coordinates to the
other are given by
22.66 X = x{ui,U2,tts)
y = I/{Mi,«2»"8)
If % and Ug are constant, then as u^ varies, the
position vector i = xi + yj + zk of F describes a
curve called the «i coordinate curve. Similarly we
define the 2*2 ^^d ^3 coordinate curves through P. The
vectors dr/auj, dt/dv^ dr/du^ represent tangent vec-
tors to the U], 112, U3 coordinate curves. Letting
•i.«2. ©3 he unit tangent vectors to these curves, we
have
Fig. 22-11
22.67
where
22.68
di . dr
dUi
h, =
dr
'*■!«
^' 31*2
"•2=2
dus
^3=3
di
dui
. fh =
dr
t ^3 ~
3r
3M3
are called scale factors. If ene^jCg are mutually perpendicular, the curvilinear coordinate system is
called orthogonaL
FORMULAS FROM VECTOR ANALYSIS
1S6
IbRMULAS INVOLVING ORTHOGONAL CURVILINEAR COORDINATES
22.69 (£r — — ^ dui + - — du2 + T — d«3 = hi dui e, + hz du^ t^ + h.^ du^ 63
OU| 0M2 "^Z
d«2 = dT'di - h{dul + hldu\ + Aj**"!
22.70
where da is the element of arc length.
If dV is the element of volume, then
22.71 dV = I (/t^ei dui) • {^262 ''"2) ^ (^^3*3 ''"a) I — '^i^'^s d«i d«2 dwa
where
22.72
SMi fltt2 flws
3(a:»!/,2)
dwi du2 dug =
3(a:, 1/, z)
a{Wi,U2,W3)
duj dz£2 dug
3(Ui, 1*2. "3)
is called the Jaeobian of the transformation.
Sx/dUi dx/dU2 dx/du^
dy/dUi dy/dii2 dy/du^
dz/dui dz/dll2 dz/du^
TRANSFORMATION OF MUlTIPlE INTEGRALS
The result 22.72 can be used to transform multiple integrals from rectangular to curvilinear
coordinates. For example, we have
22.73 (((F{x.y,z)dxdydz = rff G(u,.W2.«3) J^'^'^'^W duiduzdu^
JJJ J J J 3(Mi,M2,M3)
where ^C' is the region into which "3^ is mapped by the transformation and G(ui,W2»W3) ^^ ^h^ value of
F(», y, z) corresponding to the transformation.
0RADIENT/DIVER6ENCE, CURL ATID LAPLACIAN
In the following, * is a scalar function and A = A,«i + Agej + AgCs a vector function of orUiogonal
curvilinear coordinates Wi, «2> "a-
22 JA Gradient of * = grad * = V* = T^T^+T-|^ + r-T^
22.75 Divergence of A = div A = V-A = i-^ U^daMi) + :S7-(Mi^2) + t?-(M2^8)1
"in-afta L^**i °^2 <'**3 J
hi^i ^262 ^3*3
22.76 Curl of A = curl A = VxA = T-fr- — ~ -^
A1/1.2A3 dUi d«2 o«3
hiAi h2A.2 h^A.^
22.77 Laplacianof* = V^* = ^^^^ ^_ ^_ — j + — (^-^ — j + — (^-^ — ^J
Note that the biharmonic operator V** = V^iy^) can be obtained from 22.77.
126
FORMULAS FROM VECTOR ANALYSIS
SPECIAL ORTHOGONAL COORDINATE SYSTEMS
Cylindrical Coordinates (r, 6, z) [See Fig. 22-12]
22.78
22.79
X = rcosff, y = r sin ff, z = z
22.80
V24. ^ ^+1^1+1^+^
3r2 ^ r Sr ^ r2 5tf2 ^ dz^
Fig. 22-12. Cylindrical coordinates.
Fig. 22-13. Spherical coordinates.
Spherical Coordinates {r,e,^) [See Fig. 22-13]
22.81
22.82
22.83
r sin e cos 0, y = r sin 9 sin <p, z = r cos e
hi = 1, k2 = r2, fcg = r2 sin2 9
V24. =
r^ dr\ drJ r^ sin fl 3*
1 324,
Parabolic Cylindrical Coordinates {u,v,z)
22.84
22.85
22.86
— ^{u^ — v^), y = uv, z = z
fe| = u2 + r2, fts = 1
+
32*
dz2
The traces of the coordinate surfaces on the zy
plane are shown in Fig. 22-14. They are confocal
parabolas with a common axis.
y
u -
=^
f
11 =
i^^i.^
-^
^
^^^
i5>^
.--^
^^--<2s
Fig. 22-14
FORMULAS PROM VECTOR ANALYSIS
127
Paraboloidal Coordinates {u,v,<i>)
22.87
where
22.88
22.89
X = uv COS 0, J/ — Ml? sin 0, z =■ ^(u^ — v^)
u ^ 0, u g 0, g < 2jr
V2* =
^ ^M^i +
1 ALi*u 1-3^
«(w2 + i;2) Am I 5m y v(w2 + v2) flu I 5v / "'^'^ 9*^
Two sets of coordinate surfaces are obtained by revolving the parabolas of Fig. 22-14 about the
X axis which is then relabeled the 2 axis.
Elliptic Cylindrical Coordinates {u,v,z)
22.90
where
22.91
22.92
X = a cosh u cos V, y = a sinh u sin v, z =^ z
w ^ 0, ^ V < 27r, — CO < z < »
h\ = hi = a2(sinh2M + sin2i;), hi = X
__ 1 /a^ a^\ a^
~ a2(sinh2ii + sin^r) \Su^ &v^ ) dz^
The traces of the coordinate surfaces on the xy plane are shown in Fig. 22-15. They are con-
focal ellipses and hyperbolas.
Vff
u - 2
*1^
u = 1.
V = v
/
-a
°'>»
r
N (fr
a
1"
v =
\
\
V/
1
•
)
/
w = 2»
\
^
<;
//
"u^V^
\
^
\
J
y
/
A
u = 3/2
s
\
y
\
•a
/
A-
~^
It = 2
^
-A
\
".s
S
Fig. 22-15. Elliptic cylindrical coordinates.
128
FORMULAS FROM VECTOR ANALYSIS
Prolate Spheroidal Coordinates {$,■>], <i>)
22.93
where
22.94
22.95
X = a sinh i sin t/ cos 0, y = a sinh £ sin 1; sin ^s, z — a cosh £ cos tj
i ^ 0, ^ V ^ ^, O^0<27r
kj=zhl = a2{sinh2f + sin2,j), hi - a2 sinh2 { sin* ij
V2* =
1 5_ / . u,5*
a2(sinh2 ^ + sin^ 7?) sinh $ 5£ T'" a^
+
1
a2(sinh2 { + sin^ i?) gin ij di?
-— sin ); ^— 1 +
32*
dv J a2 sinh2 j sin2 i? S4>^
Two sets of coordinate surfaces are obtained by revolving the curves of Fig. 22-15 about the
X axis which is relabeled the 2 axis. The third set of coordinate surfaces consists of planes passing
through this axis.
Oblate Spheroidal Coordinates i$,ri,4>)
22.96 X — a cosh { cos n cos <p, y = a cosh f cos tj sin <(,, z = a sinh i sin -q
where € - 0, -7r/2 ^ ij S ;r/2, ^ < 2;r
22.97
22.98
V2* =
hi = hi = a2(sinh2£ + sin2)7), /i| - a^ cosh^ £ cos2 ,
a2(3inh2 £ + sin2 t)) cosh f 5£ l di
+
a2(sinh2 ^ + sin2 ij) cos 1? dr)
-— cos IJ -— +
32*
dv / a^ C03h2 5 cos2 ,, 302
Two sets of coordinate surfaces are obtained by revolving the curves of Fig, 22-15 about the y
axis which is relabeled the z axis. The third set of coordinate surfaces are planes passing through
this axis.
22.99
where
or
22.100
22.101
22.102
Bipolar Coordinates {u,v,z)
a sinh v
y =
a sin u
cosh V — cos u' " cosh v — cos u '
^ It < 2jr, — w < V < 09, — « < 2 < »
a;2 + (j/ — a cot w)2 = o2 csc2 u, (x — a coth ^1)2 + j/2 = *(j2 csch^ v, z = z
h-i — h2 —
(cosh V — cos u)2
, /t3 - 1
^2* = (cosh V - COS k)2 /3^ 32*\ 3^
a2 1 3u2 3^2; "^ 322
The traces of the coordinate surfaces on the xy plane are shown in Pig. 22-16 below.
FORMULAS FROM VECTOR ANALYSIS
129
y
-*•
7
^
il
ri
V V
*>
\^
/ v = -1 A
U
^
uS
(-0,0) or D = — ■
/^
s^y.-
}Xpz.
^,
*'^
.e*
(a, 0) or tf =
<\«*
Cv^^
Fig. 22-16. Bipolar coordinates.
Toroidal Coordinates {u,v,^)
22.103
_ a sinh v cos _ a sinh t> sin ^ _
cosh V — cos u cosh V — cos u
a sin u
cosh V — cos w
22.104
' ^ (cosh u — cos u)2 ' ^ (cosh v — cos u)^
,2 a^ sinh^ r
. A3 =
22
.105 V2* = (cosh u- cos w)^ d
5*
3u V cosh V — cos u du
(cosh 1; — cos m)3 _a_ / sinh v a*\ , (cosh u — cos u)^ 3f*
a^ sinh v Bv\ cosh v — cos u dv J (^ sinh^ v 3^2
The coordinate surfaces are obtained by revolving the curves of Fig. 22-16 about the y axis
which is relabeled the z axis.
Conical Coordinates (A, p., v)
22.106
(^2 _ b2)(v2 - &2)
62- a2
22.107
.2 _ , .2 _ \^{i.^ - V^) .2 _ X'(ji
X2(jf2 _ ^)
(^ - a2)(^2 - 62)
180
FORMULAS PROM VECTOR ANALYSIS
Confocal Ellipsoidal Coordinates (A,^, v)
22.108
+
+
a2-\ ' 62 - X ' c2 - X
3;='
+
yz
+
a^ — /i b^ — fi c^ — fi
x^ y2 z2
- 1 X < c2 < 62 < a2
= 1 c2<^<62<a2
= 1 c2 < 62 < P < a2
22.109
fa = (g2-x)(a2-^)(aa-.)
(a2 - 62)(a2 - c2)
2 = (62-X)(62-^)(fr2_,)
(62-a2){62-c2)
^2 = (C2-X)(c2-^)(C2-.)
(c2-a2)(c2_62)
22.110
hi =
(ti - xX" - X)
^2
A3
4(o2-x)(62-X)(c2-X)
(v - fi){\ - /i)
4(a2 - ^)(62 - /x)(c2 - n)
(X - !■)(;. - y)
4(a2-^)(62-^)(c2-^)
Confocal Paraboloidal Coordinates {k,(i,v)
22.111
'^^ 1
y2
a2-X '
62- X
^^ ,
1/2
a2 — ^
62-;,
«=^ .
y^
= z - \ — «'<X<62
- z - fi 62</i<a2
a2 - p 62 -
= Z — V
a^ < p < ^
or
22.112
^ ^ (tl2-X)(g2-;,)(a2-,)
yi =
62 -a2
(62-X){62-^)(62-p)
a2- 62
X + /I + F - a2 - 62
22.113
A? =
1.2
/12
fe^ =
(/*-
-x)(.-
X)
4{(l2-
- X){62
-X)
(.-
- n)(^ -
^>
4(a2-
-/*)(62
-l^)
(X-
-")(/*-
")
16(o2 - p)(62 - y)
DEFINITION OF A FOURIER SEtlES
The Fourier series corresponding to a function /(x) defined in the interval c ^ x ^ c + 2L where e
and L > are constants, is defined as
23.1
where
23.2
do , S / nnx , , rtirX
y + 2 (a„cos-y- + 6„sin-^
.C + 2L
nirx J
coB—=—dx
r 1 /•cfai-
If f(x) and /'(a;) are piecewise continuous and fix) is defined by periodic extension of period 2L, i.e.
f(x + 2L) = fix), then the series converges to fix) if a; is a point of continuity and to ^{fix + 0) + fix - 0)}
if a; is a point of discontinuity.
COMPLEX FORM OF FOURIER SERIES
Assuming that the series 23.1 converges to f(x), we have
23.3 fix) = 2 c„e'"'^^/^
n = — »
where
23.4 ''•' " xj fix)e-^'^^"^dx = V
^K-i6„) «>0
23.5
tSEVAL'S IDENTITY
lf'ifix))^ax = ^f^^lK^^l)
GENERALIZED PARSEVAL IDENTITY
.C + 2L
23.6 j^ J fix) gix) dx = -g- + ^2 (anC« + K^n)
c
where a„,6„ and c„,d„ are the Fourier coefficients corresponding to fix) and gix) respectively.
131
132
FOURIER SERIES
iFtheTrgW^phs
23.7
fix) =
{-:
< ar < ^
-n- < a; <
4 /sin a; . sin 3a; , sin 5a: ,
23.8 fix) = 1x1 =
X < X < TT
—X — JT < X <
£ _ A / cosx , cosSx , cos5x ,
~2t
fix)
Fig. 23-1
-2ir — n-
Fig. 23-2
277-
23.9 f(x) =X, -ir<X<Tr
2 I sinx _ sin 2a; , Bin 3x
Fig. 23-3
23.10 f{x) = X, < X < 2,7
— 2 I ^'" "^ 4- sin2x , sin3x ,
23.1 1 /(x) = Isin xl, — TT < X < n-
2 4 /cos 2x , cos4x , cos 6x
TT TT V 1-3 3*5 5-7
-2v —IT
Fig. 23-5
FOURIER SERIES
133
23.12 f(x) =
sin a; < a: < -
TT < X <2Tr
1 1 . 2 /cos 2a: , cos 4a; , cos 6a; ,
- + - sm a; - - ( -yT^ + -gTg- + -gT,- +
23.13 f(x) =
cos a; < a; < :r
— cos X —77 < X <
8 / sin 2x 2 sin 4x 3 sin 6a;
1-3 "^ 3-5 ^ 5-7
23.14 fix) =a;2, -b- < a; < tt
^ _ / cos a; _ cos2g: cos Sa;
3 \ 12 22 "^ 32
23.15 /{a;) = a;(ff-x), < a; < tt
e1 _ Aqs 2a; cos4g , cos 6a; ,
T ~ ^ 12 22 32
23.16 /(a;) - a;{ir-a;){Tr + a;), -jt < a; < tt
, „ , sin z sin 2a; sin 3g
1-^ ' ~^s~ 23 33
y^
-2r — Jf
/(a:)
r^ ^
TT 27r
Fig. 23-6
Fig. 23-9
Fig. 23-10
134
FOURIER SERIES
23.17 nx)
< a: < „-a
1 Tr — a<X<7r + a
r- + a<X<2n-
a 2 /sin a COS a; sin 2a cos 2a;
1 2
, sin 3tt cos 3a:
--)
- 2o — -- 2o -^
-» 2a -^ — 2a -
-3t -2.r — ff
" ^ 2V sir
Pig. 23-11
23.18 f(_x) =
x{ir — X) < X < r-
— a;(jr — a;) — tt < a; <
8 /sin a; sin 3a: sin 5a; ,
13
33
53
Fig. 23-12
MISCELLANEOUS FOURIER SERIES
23.19 f{x) = ain^, —v<x<v, /i # integer
2 sin /i^ / sinx _ 2 sin 2x 3 sin 3x
12 _ ^2 22 - ^2 32 - ^2
23.20 /(«) = zoBitx, —Tr<x<-tr, y.¥= integer
2^ sin P.-W I 1 , cos a: _ cos 2a; cos 3a:
2^2 12-^2 22 - /i2 ^ 32 _ ^2
23.21 /(a:) = tan-i[(aBinx)/(l - acosx)], -»■ < z < ir, |a| < 1
a sin X + -^ sin 2x + — sin 3x + • • •
23.22 /(x) " ln(l - 2acosx + a2), _^ < 3; < ^^ |a| < 1
—2 ( a cos X + — COS 2x + -^ cos 3x +
23.23 /(x) = \ tan-> [(2a sin x)/(l - a2)], -n- < x < ir, |a| < 1
a sin X + "o" sin 3x + -3- sin 5x +
FOURIER SERIES 135
23.24 f(x) = |tan-'[(2acosa;)/(l-a2)], -ir < x < -^r, \a\ < 1
a cos X — -T- cos 3x + -=- cos 5x —
S
23.25 fix) = e>^, -^ < x < w
2 sinh /tiT ( \_ , ■^ (— l)"(/t cos nx — n sin na:)
23.26 f{x) = sinh /lar, -b- < a; < n-
2 sinh /in- / sinx _ 2 ain 2x 3 sin Sa:
12 + ^2 22 + u2 "^ 32 + u2
23.27 /(x) = cosh iix, -tt < x < n
2fi sinh fiTT /_1 cos a: cos 2g _ cos 3g ,
TT I 2/x2 12 -|_ ^2 "^ 22 + /i2 32 + ^2
23.28 /(x) - In jsin^x], < a; < n-
-(1„2+ c^+C0|2£^C0|3x^
23.29 fix] = ln|cos^K|, -w < x < v
_ f 1 2 — <^osg COS 2a: _ cos 3x ,
23.30 fix) = ^5r2 - ^irx + ^2:2, ^ a: ^ 2)r
cos a: cos 2a: , cos3g ,
12 "*■ 22 32
23.31 /(x) ^ ^X(X - n){x - 2;r), g X S 2;r
sinx sin2x ainSx ,
13 ■•" 23 33
23.32 fix) = ^^^ - ^w^^ + ^7rx3 - ix4, ^ X S 2.
cos a; cos2x cos3x ,
14 "*■ 24 3^
^^^^^^^^^^^^^^^^ BESSEl'S DIFFERENTIAl EQUATION
24.1 x^y" + xy' + {x^-nZ)y = n^O
Solutions of this equation are called Bessel functions of order n.
BESSEL FUNCTIONS OF THE FIRST KIND OF ORDER n
24.2
24.3
24.4
Jn(x) =
X"
1 -
+
2«r(n + l)l 2(2m + 2) 2 • 4{2n + 2)(2n + 4)
^ "I (-l)''(a;/2)" + 2fe
J-nM = T^z:
1 -
+
X^
2-n r(i _ „) Y 2(2 - 2n) 2 • 4(2 - 2n)(4 - 2n)
^ (-l)fc(g/2)2fe-"
k=o /c ! r(fc + 1 - ?i)
J_„(x) = i-l)^J„(x) n = 0,1,2....
If n#0,l,2,..., y„(x) and J_„(x) are linearly independent.
If 717^0,1,2,..., J„ (a:) is bounded at x =^ while ./_„(x) is unbounded.
For n = 0, 1 we have
24.5 Jo{x) — 1 — 22 + 22. 42 ~ 2^ * 4^ * 6^ + " ' '
_ . , - , . _ » _ g^ , ac^ x[
24.0 Ji(a;) - ^ 22 • 4 "*" 22 • 42 • 6 22 • 42 ■ 6^ • 8
24.7 /o'C'^) = -M'^)
BESSEL FUNCTIOI
^KIND OF ORDER n
24.8
i'nla:) =
/„(«;) coswn- — ./-n(a:)
sinTtTT
Jp(x) cospn- - ./-p(a:)
lim -i-
M9^0,l,2, ..
n = 0,1,2, ..
This is also called Weber's function or Neumann's function [aUo denoted by iV„(a;)].
1S6
BESSEL FUNCTIONS
137
For n = 0, 1, 2, . . . , L'Hospital's rule yields
24.9 Y^ix) = -{ln{x/2) + y}J„{x) -- '^ in-k-l)\ix/2)^k-n
T IT fc=o
--2 (-iH-i>(fe) + f(n+fc)} 7;'^^
^ fc=o kl{n + k)\
where y = .5772156. , . is Euler's constant [page 1] and
24.10 «i.(p) = 1 + 1+ 1+ ••• +-, <i.(0) = ■
For n = 0,
24.11 Y,{x) = |on(x/2) + r}Jo(x) + f|f-^(l + i) + 2^{l + i + i)
24.12 r_„(x) = (-l)''Y„(x) n = 0,1,2,...
For any value ti £ 0, J nix) is bounded at a: = while 3'„(a;) is unbounded.
24.13 y = AJJx) + BJ.^ix)
24.14 y = AJJx) + BY„[x)
n-^ 0,1,2,
all n
/ax
where A and B are arbitrary constants.
GENERATING FUNCTION FOR 3n{z)
24.16
gX(t
-i/t)/2 = 2 J„(a;)(«
24.17
24.18
24.19
24.20
RECURRENCE FORMULAS FOR BESSEL FUNCTIONS
■'n + l(«) = —•'«(«) - •^n-l(a;)
a;J4{a!) ^ a;/„_j(x) — nJ„{a;)
a;J^(a;) = nJ'„(ir) - xJ^^-^{x)
d
dx
24.21
24.22
The functions Y„{x) satisfy identical relations.
{x»J„(a:)) = os»J^-i{x)
^{x-«J^{x)} = -x-r'J^^.ix)
138
BESSEL FUNCTIONS
BESSEt FUNCTIONS OF ORDER EQUAL TO HALF AN ODD INTEGER
In this case the functions are expressible in terms of sines and cosines.
24.23 Ju2(x) = \-^sin!t
2 .
24.26 J_,Mx) = -W-^
2 /cos X
WX \ X
+ sin a:
24.24 J_,/2(a:) = V— cos*
vx
24.27 75/2(3
— 1 1 sin X — — cos X
X
24.25 J,„(x) = ylJ(^^-cosx^ 24.28 J.,„M - ^/^jf sinx + (|- 1
For further results use the recurrence formula. Results for yi/2(«). Y^/si^), . . . are obtained from 24.8
HANKEL FUNCTIONS OF FIRST AND SECOND KINDS OF ORDER n
24.29 Hl,'\x) - J,ix) + iY,{x)
24.30 Hl^'ix) - J«(x) -ir„(x)
BESSEL'S MODIFIED DIFFERENTIAL EQUATION
24.31 . x^" + xy' - {x^ + n^)v = n^O
Solutions of this equation are called modified Bessel functions of order n.
lED BESSEL FUNCTIONS OF Tl
24.32 /„(») = i-^'JJix) = e-^^!^J^(ix)
JT"
1 +
+
xi
2"r(« + l)r 2(2?i + 2) 2 • 4{2n + 2)(2n + 4)
KIND OF ORD1
+ -■ - 2
24.33
/_„(x) = i"J-„(ix) = e^^'^J.^iix)
a;-n
1 +
+
X4
+
2-«ra-n)\ 2{2-2n) 2*4{2-2n)(4-2n)
24.34 I-^{x) = 7„(x) n = 0,l,2,...
If n -/■ 0,1,2, .. ., then /„(«) and /_„(a!:) are linearly independent.
For n = 0, 1, we have
^ 4. ^ A- »°
22 "^ 22>42 "*" 22 '42. 62
24.35 /o(x) = 1 + 22 + 22712 + 02. d2.,-- +
•9A '«A r / ^ — £ J. "^ 4. g° , x'' ,
^*».-*o 7i(a;j - 2 "*" 22.4 22 -42 '6 2^ • 42 • 62 • 8
{a:/2)
N + 2h
ft=ofc!r(n + Jfc+l)
= 2
(a:/2)
2fc-«
fc=o k\T{k + l-n)
24.37 /^(x) - /i(x)
BESSEL FUNCTIONS
139
MODIFIED BESSEL FUNCTIONS OF THE SECOND KIND OF ORDER n
24.38 K„(x) =
—^— {/_„(x) - /„{x)} « ^ 0, 1, 2, . . .
2 sin njT
Urn
{/_„(x)-/»} n = 0,1,2,...
p^n 2 sin Pit '' ^
For n = 0, 1, 2, . . . , L'Hospitars rule yields
24.39 KJ,x) = (-1)"+ Hln (a:/2) + 7>/„(a;) + J- 2 ("DM^ - A: - 1) ! (a:/2)2k-«
«-i
2 2
+
2 k^ofc!(« + ft)!
{*{fc) + ^(n + fc)}
where *(p) is given by 24.10.
For n = 0,
24.40 ifoCx) = -{In (x/2) + y}/o(«) + ^ + 3^742(1 + *> + gg.^g.ga (1 + i + i) + " ' *
24.41 K-J?c) = K^{x) 71 = 0,1,2,...
X*
GENERAL SOLUTION OF BESSEL'S MODIFIED EQUATION
24.42
24.43
24.44
where A and B are arbitrary constants.
V = Al^ix) + Bl.^ix)
n^0,l,2
y = AI„(x) + BK^ix)
all n
y = Al^ix) + BI^(x) f-^
^ xl„{x)
alln
GENERATING FUNCTION FOR /n(x)
24.45
ei(t + i/«/2 = 2 /„(a:)t"
RECURRENCE FORMULAS FOR MODIFIED BESSEL FUNCTIONS
0-,
24.46 /,+ ,(x) = I„-i(x) ~ —I„(x)
24.47 lUx) = ^{/„„,(x) + /„+i(x)}
24.48 x/;(a!) = xl^-iix) - nl„(x)
24.49 xl'„(x) = xl„^i{x) + nl„(x)
24.50 ^{x^Ux)} - x"/«_i(a;)
277
24.52 K^+.ix) = A-„_i(x) + — K«(ar)
24.53 <(a;) = ^iK,- lix) + K^^t(x)}
24.54 a:<(x) = -xK^.^ix) - nK^ix)
24.55 xK;(x) = ni:„(x) - xK„+i(«)
24.56 ^{x"£:„(x)} = -x-iC„_i(a:)
24.51 ^{a:-n/,(x)} = x-«7„ + i(x)
24.57 ^{x-"iC„{x)} = -x-"K,+ ,(x)
140
BESSEL FUNCTIONS
MODIFIED BESSEL FUNCTIONS OF ORDER EQUAL TO HALF AN ODD INTEGER
In this case the functions are expressible in terms of hyperbolic sines and cosines.
24.58 /i/2(x) = -\|— sinha;
24.59 /_i/2(a:) = A/4co8ha:
24.61 /_3,,(x) = ^(^sinh.-^
24.62 /5/2(a:) - a/— "1(4+0 sinhx - - cosh a;
24.60 /3/2(x) - A/-±-fcoshx-
sinha;
•\ / — 1 ( ~9 + 1 1 co3^ ^ sinh X >
24.63 /_5/2(x) = ^/^■{(^+ 11 cosh:
For further results use the recurrence formula 24.46. Results for K^/^i.^), K^/2{x), ... are obtained
from 24.38.
Ber AND Bei FUNCTIONS
The real and imaginary parts of Jjt (aJe3»ri/4) are denoted by Ber„ (a;) and Bei„ (x) where
OAAA x» / ^ -V (a;/2p + « (3n + 2fe)rr
24.65
If « = 0,
24.66
24.67
r, ■ / \ X (a;/2)2fc + " . (3n + 2fc):r
Ber(a:) - 1 " ^72" + -4T2 •'
Bei (x) = (a:/2)2 - -i-g^ + ^,^
Ker AND Kei FUNCTIONS
The real and imaginary parts of e~'^'^^^ KJ^{xe^''^) are denoted by Ker„ (a;) and Kei„ (x) where
24.68
Ker„ (a:) = -{In (a;/2) + y} Ber„ (x) + ^^ Bei„ (a:)
(k - fc - 1) ! (^/2)2fc-" (3n + 2A:);r
r-, cos ;;
k\ 4
1 S^ (a;/2)«+2fc
(3n + 2kW
4
24.69
Kei„ (a:) = -{In (x/2) + y) Bei„ {x) - J,r Ber„ (x)
n-l
(TO-A:-l)!(a;/2p-« . (Sti + 2fc)7r
. {3n + 2fc)7r
_^ 1 ^ a;/2« + 2ic , ^, _^ , „ . (3n +
and * is given by 24.10, page 137.
If 71 3:: 0,
(a;/2)''
(a;/2)8
24.70 Ker (x) = -{In (x/2) + y} Ber (x) 4- jBei (x) + 1 - -^^{1 + ^) + ^^(1 + ^ + ^ + J) -
24.71 Kei (X) = -{In (x/2) + y} Bei (x) - ^ Ber (x) + (x/2)2 - ^^ d + ^ + i) +
BESSEL FUNCTIONS
141
DIFFEI
"fOR Befe^fKer, Kei FUNCTfONS
24.72
kV + xy' ~~ {ix^ + v?)y =
The general solution of this equation is
24.73 y = A{Ber„ (a;) + i Bei„ (x)} + B{Ker„ [x) + i Kei„ (a:)}
GRAPHS OF BESSEL rUNCTIONS
Fig. 24-1
Fig. 24-2
1— a;
Fig. 24-3
Fig. 24-4
Beia
Kei X
Fig. 24-5
Fig. 24-6
142
BESSEL FUNCTIONS
INDEFINITE INTEGRALS INVOLVING BESSEL FUNCTIONS
24.74 C xJQ(x)dx = zJi {x)
24.75 C x^Joix) dx = x^Jiix) + xJo(x) - f Joix) dx
24.76 Cx-^jQ{x)dx = x'^Jiix) + (m-l)x'"-^ Jo(x) - (m-l)2 f x'»-2 ^(a;)
dx
M^)
Mx)
)dx
24.77
24 78 f ^ -
J„(x)
dx =
Jy{x)
Joix)
-1 (?n-l)2j
Jai'^)
dx
24.79 r^i(a;)da = -Jo{x)
24.80 ra:yi(a:)da; = -xJo(x)+('jQix)dx
24.81 ra;'"Ji(a:)da: = -x« /„ (*) + *» f ^"""^ *'o(») (^a^
24.82 r^^dx = -J, (x) + J Jo (a') da:
y,(x)
24.83 f-'-l^dx ^ -Jl^^^^r-^dx
J x^ 7nx"*~^ m ./ a;*" *
24.84 fx^Jn-iix)dx = x^Jnix)
24.85 ( x-^J„+i(x)dx = -x-^J„{x)
C x'^J„{x)dx = -x'^J„-i(x) + (m + n-1) J z"""' J„_i(x) d«
24.86
24.87
C xJn{ax)Jn(px) dx =
x{a JJPX) Kiax) - /3 J^(aX) J^M)
)S2~ a2
^2 / «2
24.88 Ja;/2(«x)dx = |- V;(««)}2 + y (^1 " ^j(-'n('«=^)P
The above results also hold if we replace J„{x) by Y„{x) or, more generally, A y„(x) + B y„(a!) where
A and £ are constants.
DEFINITE INTEGRAiS INVOLVING BESSEL FUNCTIONS
24.89 Ce-'"Mbx)dx =
24.90 C e-'"J^{bx)dx =
24.91 I cos ax Jo(bx) dx
Va2 + &2
(Vgg + 62 - a)n
6"Va2 + 62
1
Va2-62
n > -1
a> 6
a< 6
BESSEL FUNCTIONS
143
24.92 C J„ibz)dx =1 n>~l
24.93 J^ ^^^^ = i n = l,2,3,...
24.94 r e-'^jQ{by/^)dx = ?lU^
Jq a
24.95 I X Jn(ax) J^ifix) dx =
i82-a2
24.96 C xJliax)dx = ^U;(«))2 + ^{l-n2/«2){Jn(«)P
J a:yo(o3;)/o(j3fl:) (ix = ^^^
24.97
24.98 yo(a:)
24.99 ^.(x)
24.100 ^(x)
24.101 Yo(x)
24.102 /o{:c)
^2
INTEGRAL REPRESENTATIONS FOR BESSEL FUNCTIONS
= - I cos {x sin 9) d9
= — I cos (n* — ac ain tf) d*, n = integer
= _ I COB (a; giQ g\ cos^ tf dtf, Tt > —4
2 r"
= I COS (a: cosh w) du
- I cosh {x sin e) d$ = ~ \ e* ^me d*
ttJ,, 2:rJ„
ASYMPTOTIC EXPANSIONS
24.103 J„(rr)
24.104 Y,{x)
24.105 J,(:r)
24.106 Y^{x)
24.107 /„(»)
24.108 £:„(ie)
2 / na- _
COS ar — s" ~ T
jra; \ 2 4
2 . / nr- jr
Sin I X ~
TTX \ 2
'2s^ \2.nJ
2_! ex
n-n V 2n
V2^
where z is large
where x is large
where n is large
where n is large
where x is large
where x is large
144
BESSEL FUNCTIONS
ORTHOGONAL SERIES OF BESSEL FUNCTIONS
Let Xi, \2> M,--- be the positive roots of R J„ (x) + Sx J^(x) = 0, n> -1. Then the following aeries
expansions hold under the conditions indicated.
S ~0, R^'O, i.e. Ai, A2, As, . . . are positive roots of Jn(x) =
24.109
where
24.110
fix) = A^J^iXix) + A2Jni\2^) + A^J^iX^x) +
2 C^
In particular if n. = 0,
where
24.112
2 r'
^fc ^ 72777 I a;/(a;)^o(M)d:a'
RIS > -«
24.113
where
24.114
/(jr) = A^J^{\^x) + A2J„(X2a;) + A3J„(X3a:) +
A^ =
J */(a:).^n('^fca;)dx
In particular if m = 0,
where
R/S = -n
24.117
where
24.118
24.119
where
24.120
fix) = AqX» + AiJ^{\ix) + AzJniH'^) + •••
Aq = 2(»+l) r x^+^fix)dx
"^^ " j''(x)-j ^xw — ;r; J <^m'^niK^)dx
^ •'n(Xfc) — J„_i(Xfc) J„ + i{Xk) -'o
In particular if n = so that i2 = [i.e. Xi, Xg, X3, . . . are the positive roots of ^i (a;) = 0],
fix) = Ao + AiJoiXiX) + AzJoiXzx) + ••■
Ao = 2J xf(x)
dx
A^ =
4iw
J X fix) J^iX^x) dz
BESSEL FUNCTIONS
145
R/S < -n
In this case there are two pure imaginary roota —iKq as well as the positive roots \i, \2> H'
and we have
24.121
where
24.122
fix) = Ao/„{Xox) + AjJ„(Xia:) + AsJniM^) + •"
j X fix) J„(\kX) dx
A. =
Jni\k) - Jn-liK)Jn-niW ^0
M^^UANEOUS^^^^
24.123 cos (a; sin «) = Jo{x) 4- 2Ji{x)cos2e + 2Jt{x)cos4e + ■•■
24.124 sin{a;sinfl) = 2Ji(x)ame + 2J3(x)sin3ff + 2J5(x)sin5tf + ■••
24.125 Jni^ + y) = 2 Ji,ix)Jr,-k{y) n = 0, ±1,^2, . ..
fc= -"
This is called the addition formula for Bessel functions.
24.126 1 = Jo{x) + 2J2{x) + ■■• + 2/2«(«) + ■"
24.127 X = 2{Mx) + ZJsix) + 6J5(x) + •■■ + (2n+l)J2„+i(x) + ••■)
24.128 a:2 = 2{4J2ix) + ieJ^(x) + 36Je(a:) + •■■ + (2n)2/2„(a;) + •••}
X J I (x)
24.129
= J^ix) - 2Mx) + SJ^ix) - ••■
24.130 1 = 4(x) + 2Jlix) + 2JI(x) + 2j|(ar) + •••
24.131 J'^ix) = i{J„-2(x) - 2J„{x) + ^„ + 2(«)>
24.132 J'^'ix) = iU„-8(«) - 3J„_x(ac) + 3J„ + j(x) - Jn + 3(a:)}
Formulas 24.131 and 24.132 can be generalized.
24.133 K{x)j^^ix)-JL,JM = ^''""'
vx
24.134 y„(x)J_„ + i(x) + y-„(a:)J„_i(x) = ^ "" "'^
24.135 J„^,{x)Y„{x) - JAx)Y,^tix) = -
24.136 sin a; = 2{J,(x) - J^ix) + ^{x) - •■•}
24.137 cosx = Jo(x) - 2J2(a=) + 2^4(2:) - ••■
24. 1 38 sinh x =^ 2{/,(x) + h{x) + h{x) + • ■ ■}
24.139 cosh x = /©(x) + 2{/2(x) + I^ix) + h(x) + • • •>
LEGENDRE'S DIFFERENTIAL EQUATION
25.1 (i^j^2)y" _ 2xy' + n(n + l)y =
Solutions of this equation are called Legendre functions of order n.
LEGENDRE POLYNOMIALS
If m — 0, 1, 2, . . ., solutions of 25.1 are Legendre polynomials Pn(a:) given by Rodrigue'a formula
25.2
^'^(=^> = 24!£^(^^-^>"
^
25.3 P^ix)
25.4 P,{x)
25.5 P^{x)
25.6 P^ix)
^ 1
1(3x2-1)
^(5x3 - 3x)
SPECIAL LEGENDRE POLYNOMIAL
25.7 P^{x) = ^(35x^-30x2+3)
25.8 P^ix) = ^(63x5 - 70x3 + 15a;)
25.9 Ps(x) = Jg{231a:8-315x-» + 105x2-5)
25.10 Pj(x) = J^(429x7- 693x5 + 315x3 -35x)
LEGENDRE POLYNOMIALS IN TERMS Of ^ WHER!
25.11 Fo(cos#)
25.12 P,(cos(?)
25.13 /»2(costf)
25.17 Pe(coss)
25.18 PyCcoss)
1 25.14 P3(cos«) = ^(3 cos tf + 5 cos 3?)
costf 25.15 P4(costf) = JL(9 + 20cos2ff + 35 cos4ff)
J(l + 3 cos 29) 25.16 Pstcos 8) = i|8(30 cos « + 35 cos 3a + 63 cos 59)
_ 1
~ ^(50 + 105 cos2ff 4- 126 cos4ff + 231 cos 6ff)
- 15^(1 "^B cos 9 + 189 cos 3a + 231 cos 5# + 429 cos 7«)
25.19
GENERATING FUNCTION FOR LEGENDRE POLYNOMI
= 2 Pn(^)t"
Vl - 2tx + (2 n=0
146
LEGENDRE FUNCTIONS
147
25.20
25.21
25.22
25.23
25.24
URRENCE FORMULAS FOR LEGENDRE POLYNOMI
(n + l)P„+,(a!) - (27i+l)xi>„(x) + nP^_,{x)
F; + i(x) - xP:(x) = (n+l)P„(x)
xP;(x) - P'„-iix) = nP„(x)
K + A') - P'n-iix) = (2n+l)P„ix)
(x^~l)P;,{x) = nxP^(x) ~ nP^_,(x)
^
25.25
25.26
OLYNOMI
P^{x)P^{x)dx = m^n
r,
J_ {Pnix)}
^dx =
2n+ 1
Because of 25.25, Fm(«) and P„ix) are called orthogonal in -1 ^ a; ^ 1.
25.27
where
25.28
jOGONAl SERIES OF LEGENDRE POLYNO
/(x) ^ A^P^ix) + A,P,(x) + A^Pzix) + ■■-
2k +1 C^
A^ -
J mP^(x)dz
iSPECIAL RESULTS INVOLVING LEGENDRE POLYNOMI
25.29 P„(i) = I
25.32
25.33
25.34
25.35
25.36
25.30 P„(-i) = (-i)t
PniO) = i
25.31 P„(-x) = (-l)"P„(x)
n odd
(-1)
n/2
1 ■ 3 ■ 5 • ■ • (71 - 1)
n even
1 /"^
''nCa;) = - I (« + Vk^-I cos 0)" d^
/^n(a:
)da; =
^«+i{»)-P„-i(x)
2n + l
-"W = 2^£#^''^
where C is a simple closed curve having x as interior point
148
LEGENDRE FUNCTIONS
GENERAL SOLUTION OF LEGGNDRE'S EQUATION
The general solution of Legendre's equation 13
25-37 y = A[7„(x) + BV^{x)
where
25.38 U ix) = 1 - ^^^^±11x2 + "(»-2)(^ + l)(^ + 3) _ ^ ,
n\ / 2! 4!
25.39 VM = X- t"-l)(^ + 2) (n-l)(n-3)(n + 2)(« + 4)
n\ / 3! SI *
These series converge for — 1 < x < 1.
LEGENDRE FUNCTIONS OF THE SECOND KIND
25.40
where
25.41
If n = 0, 1,2, . . . one of the series 25.38, 25.39 terminates. In such cases,
C/„(x)/f;„(l) n = 0,2,4,...
'f'n(«)/^n(l) »i -1,3,5,...
n2 /
C;„(l) = (-l)"/2 2n
ir-
n\
n = 0,2,4, ...
25.42
^n{l) = (-l)<''-l)/2 2"-»
n-1
nl n= 1,3,5,...
The nonterminating series in such case with a suitable multiplicative constant is denoted by Q„(x) and
is called Legendre's furiction of the second kind of order n. We define
25.43
Qnix) =
' U„{l)V„ix) n = 0,2,4,...
~VM)U„ix) M-1,3,5,...
25.44
25.45
25.46
lAl LEGENDRE FUNCTIONS OF THE SECOND KIND
Q.(x) = |ln(i±f
<?'<^> = p4\H^-^
nu\ - 3x2-1 / l + x \ 3x
25.47
i^\ - 5a3-3x, f l + x \ 5a:2 , 2
The functions Q„(fl:) satisfy recurrence formulas exactly analogous to 25.20 through 25.24.
Using these, the general solution of Legendre's equation can also be written
25.48
y = AP^ix) + BQ^ix)
26.1
LEGENORE'S ASSOCIATED DIFFERENTIAL EQUATION
(1 - x'^)y" - 2xy' + ^ n(n + 1) -
l-a;2
y ^
Solutions of this equation are called associated Legendre functions. We restrict ourselves to the im-
portant case where m, n are nonnegative integers.
■GENDRE FUNCTIONS Qd
26.2
rim
P^(X) = (1- 0:2)^/2 ^P„(X) =
(1 _ x2)m/2 (fm+n
where Pni^) are Legendre polynomials [page 146]. We have
26.3 Plix) = PJx)
-(a.2-1).
26.4
P^{x) = if m > n
SPECIAL ASSOCIATED LEGENDRE FUNCTIONS OF THI F|RST KIND
26.5 Plix) = (l-a;2)i/2
26.6 PUx) = 3a;(l-x2)i/2
26.7 PUx) = 3(1 -ir2)
26.8 Plix)
26.9 Plix)
26.10 Plix)
|(5a;2-l)(l-a;2)i/2
15a;(l-x2)
15(l-a;2)3/2
26.11
GENERATING FUNCTIO^O^S
(2m)!(l-gg)"'^gt"
2 i'r(a')i"
n=in
26.12
26.13
RECURRENCE FORMUL
{n + l-m)P'^+iix) - (2n + l)xP';:(x) + in + 7n)P^-i{x) =
Pr\^)-W^^zPr\^) + (n-m)in + m + l)P:ix) =
(1 - a;2)i
149
150
ASSOCIATED LEGENDRE FUNCTIONS
26.14
26.15
f P™{a;)Pr{x)dx = if n^Z
'•'-1
)}2da! =
2 (m + m) !
2n + 1 (rt - Tn) !
26.16
where
26.17
THOGONAL SERIES
fix) = A^PZix) + A^^.PZ^iix) + A^^^Pl+^M +
A. =
2k + 1 {k-m)]
^ - ~l~(itT^ J_/<^'^'=t^''^^
i£.
ASSOCIATED LEGENDRE FUNCTIONS OF THE SECOND KIND
26.18
(fm
where Q„{x) are Legendre functions of the second kind [page 148].
These functions are unbounded at a: = ±1, whereas P^(x) are bounded at a: — ±1.
The functions Q^ix) satisfy the same recurrence relations as P!^{x) [see 26.12 and 26.13].
GENEJ
IF LEGENDRE'S ASSOCJ
kOUATION
26.19
V = AP:^(x) + BQr(a')
17
HERMrTE POLYNOMIALS
27.1
NTIAI EQUATI
y" — 2xy' + 2ny =
If n = 0, 1, 2, . .
Rodrigue's formula
27,2
RMITE POlYNOMIAtS
then solutions of Hermite's equation are Hermite polynomials H^(x) given by
H^x) = (-i)nex'£L(g-^^j
27.3
Hoix) = 1
27.4
Hi(x) = 2x
27.5
Hsix) = 4a;2 - 2
27.6
Hsix) = Sx^- 12a;
P6CIAI HERllTri»OlYNOMIALS
27.7 Hi(x) = 16a;-i - 48z^ + 12
27.8 H^{x) = B2z^ - 160x3 + i20x
27.9 H^ix) = 64fl:8 _ 480x4 + 720x2 - 120
27.10 H^(x) = 128x' - 1344x5 + 3360x3 - 1680^
27.11
OENERAtiNG FUNCTION
,2tz-t' ^ -V ^n(a^) *"
nio nl
27.12
27.13
EECURRENCE FORMUL
■«n+i('K) = 2xff„(x) - 2nH„_i(x)
Hl,ix) = 2nH„_j{z)
151
152
HERMITE POLYNOMIALS
27.14
27.15
ORTHOGONALITY OF HERMITE POLYNOMIALS
r 6-==* H^{x) Hnix) dx = m¥=n
27.16
where
27.17
ORTHOGONAL SERIES
fix) = A^H^ix) + A,H,{x) + A^H^ix) +
A^ - ^= f e'^' f{x)Ht,ix)dx
27.18
SPECIAL RESULTS
BJ,) = (2x)n - 2l(ZL^(2.)n-:. + ^(^- l)(^^;2)(--3) (2.)n-4 ,
27.19 H„(-a;) = (-l)''fl„(a:) 27.20 H2„_,(0) =
■ff2n(0) = (-1)''2« • 1 • 3 • 5 ■ ■ • (2m,- 1)
«„ + ,(«) H„+i{0)
27.21
27.22
27.23
27.24
27.25
J ^«(t)
±f.-^
dt =
2(n + 1) 2{n + 1)
,-!*
dx
{e-'^H^ix)} = -e-^fl„ + i(a;)
J ("e"** Hnixt) dt = Vvnl P„(a;)
27.26 ^ i /n
fc=0
This is called the addition formula for Hermite polynomials.
HJ^x^y) = 2 ^j^(l]H^{xyf2.)H^.^(yyf2)
27.27
" H^{x)H^{y) _ g„+,(ir)H„(y) - H„(x)H^+,iy)
2" + %! (a — 1/)
28.1
xy" + {X-x)y' + ny =
lAOUERRE POLYNOMIALS
Zl7J:\-l:J. '*"'" '°'''**°"' '' Laguerre's equation are Laguerre polynomials L„ix) and
by Rodrigue'a formula.
28.2
are given
rf«
^» = ^:^i^"'-')
da:"
28.3
Lo(x)
= 1
28.4
L,{x)
= ~x+l
28.5
L^ix)
= 3:2 - 4a; + 2
28.9
L^{x)
= x6 - 36*5 +
28.10
Liix)
= -a;7 + 49a;fl
SPECIAL LA6UERRE POLYNOMIAL!
28.6 Lsix) = ~3^ + 9x2 - 18a; + 6
28.7 ^^(a;) ~ x^- 16x3 + 72^2 - 96a; + 24
28.8 Lsix) = -x^ + 25x4 - 200xS + 600x2 - 600« + 120
x6 - 36x5 4- 450x4 _ 2400x3 + 5400x2 - 4320x + 720
-x7 + 49xfl - 882a:5 + 7360x4 - 29.400x3 + 52,920x2 - 36,280x + 5040
28.11
GENERATING FUNCTION
- L„{x) <n
e-xtn~t
1 - t
28.12
28.13
28.14
K+iix) - (2n + 1 - x) L„(x) + m2L„_,(x) =
lUx) ~ nLl_i(x) + nL„_^(x) =
a!Z*^(x) = nL^ix) - n2L„_i(x)
153
154
LAGUERRE POLYNOMIALS
28.15
28.16
MlAlS
28.17
where
28.18
ORTHOGONAL SERIES
f(x) = AoL(,(x) + A^Liix) + A^Lzix) +
1 r°*
28.19 L„(0) = m!
28.21
28.22
28.23
28.24
28.25
28.20 J L,{t)dt = L„ix) - -^^^^Y
II
2!
I a;Pe-^L„(x) da; := -^
''o [(-l)«(m!
(-!)««!
if p < n
)2 if p = n
(n\)^{x~y)
L„(a;) = r «"«*-« Jo (2-/^)
d«.
lAGUERRE'S ASSOCIATED DIFFERENTIAL EQUATION
29.1
xy" + (m + 1 — x)y' + {n — m)y =
iSOCIATED LAGUERRE POlYN<
Solutions of 29.1 for nonnegative integers m and n are given by the associated Laguerre polynomials
29.2
where Ln(a;) are Laguerre polynomials [see page 153],
29.3
29.4
Lnix) = Ln(x)
L^{x) = 9 if m > n
ISSOCIATED LAGUERRE POLYNOMIAI.
29.S
l\{x) = -1
29.10
Llix) - -6
29.6
Llix) = 2x-4
29.11
Llix) = 4a;3 - 48a:2 + UAx -
-96
29.7
Llix) = 2
29.12
lI(x) = 12«2 - 96a; + 144
29.8
Llix) = -3a:2 + 18a: -
-18
29.13
Llix) = 24a; -96
29.9
Llix) = ~Gx + 18
29.14
Ltix) = 24
GENERATING FUNCTION FOR L^j
29.15
(1 -«)" + !
- Lnix)
2 -r^***
155
156
ASSOCIATED LAGUERRE POLYNOMIALS
ORMULi
29.16
r-i — Ln + i(x) + {x + m — 2ti—l)Lnix) + n^Lrt-iix) =
71 -j- 1
29.17
■^{Ln{x)) = Ln (X)
29.18
■^{x-^e-'^ Ln{x)) = (m~n-l)a:'"-'c-^Ln \x)
29.19
x^{Ln{x)} - ix — m)Ln(x) + {m — n — l)Ln (x)
ORTHOGONALITY
29.20
{c««-* Ln(x) Lpix) dx = p¥^n
A
29.21
X
^ x-e-HLn(xWdx = ^^
ORTHOGONAL SERIES
29.22
where
29.23
f(x) = A^Lm(x) + A^+iLm-nix) + A^ + 2^m+2{x) +
A^ =
{k~
{k\)
x^e~^Lh (x) fix) dx
^SPECIAL RESULTS
29.24
-m, . ,, m! nln — m} , n(n ~ l){n ~ m){n ■- m — 1)
^nix) = (-l)«^^-^|x«-".- -L_^a:«-'«-l + -^ '.^^-^ L-^n-^--
, 3;n— m — 2 ^
}
29.25
r" «. + , rrr^'/M^j (2«-Tn + l)(K!)3
CHEBYSHEV'S DIFFERENTIAL EQUATION
30.1
(l-x2)y" ~ xy' + n'^y = n = 0,l,2,...
CHEBYSHEV POLYNOMIALS OF TH£ FIRST KIND
Solutions of 30.1 are given by
30.2 r„(x) = cos (71 cos-i a!) = a:«
- C"^x"-2a-a:2) + /'"'\a;"-» (1-052)2 _
SPECIAL CHEBYSHEV POLYNOMIALS OF THE FIRST KINI
8x* - 8fl;2 + 1
16ar5 - 20x3 + 6a;
32x« - A8x* + 18x2 _ 1
64a:' - 112z^ + 56x3 _ ij^
30.3
To(x) = 1
30.7
T^ix)
30.4
T,{x) = X
30.8
Ts(x)
30.5
Tzix) = 2a;2-l
30.9
Tsix)
30.6
Tsix) = 4x^-Sx
30.10
Tj{x)
GENERATING FUNCTION FOR Tn{x)
30.11
I- tx
1-Ztx+ t^
2 r„(x)t«
n=0
SPECIAL VALUE!
30.12 T^{-x) = (-l)T„(a;)
30.13 ^^{l) = 1
30.14 r„C-l) = (-1)"
30.15 r2^(0) = (-1)«
30.16 r2,+,(0) -
157
158
CHEBYSHEV POLYNOMIALS
Fira^^rFORMuraFo
30.17
T„^i{x) - 2xr„(x) + r„_i(a:) =
30.18
30.19
/
I T^{x)T„ix)
J
-1 Vl-3:2
1 {r„(x)}2
da; = m ¥= n
dx =
TT if n =
jr/2 if n= 1,2, .
30.20
where
30.21
fix) = \A^T^{x) + A,T^(x) ■\- A^T^ix) +
2 r fix)T^{x}
-IS
-1 \/l-x2
dz
30.22
IHEBYSHEV POIYNOMIALS OF TJ
;OND KIND
Un(x) =
sin {(n + 1) cos~^ a:}
sin (cos""^ a;)
n+ 1
a;" —
n + 1
g xn-2Cl-x2) + , 5
m + 1
.11-4/1 —^212 _
x"-4(l-a:2)
SPECIAL CHEBYSHEV POLYNOMIALS OF THE SECOND KIND
30.23
Vcix) = 1
30.27
Uiix) '-
= IQx* - 12a;2 + i
30.24
Uiix) ^ 2z
30.28
U^x) --
= 32x5 _ 32a;3 + 6x
30.25
Uzix) = 4a;3-
- 1
30.29
U^ix) =
= 64x8 _ 80x4 + 24x2 _ i
30.26
U^ix) - 8x3-
4x
30.30
Ujix) --
= 128x7 _ i92a;5 + gOx^ - 8x
FUNCTIOI
30.31
1 - 2tx + t2
2 u^{^)tn
n=0
CHEBYSHEV POLYNOMIALS
169
30.32 U„(-x) = (-l)«t/„(x)
30.33 UJl) = n+1
30.34 I7J-1) = (-l)«(n+l)
30.35 U2n(0) = (-1)«
30.36 [72,,, (0) =
30.37
RECURSION FORMULA FOR Un{x)
U^^^ix) - 2xU„(x) + t7„_i(x) =
ORTHOGONALITY
30.38
30.39
30.40
where
30.41
30.42
30.43
30.44
30.45
30.46
f Vl-a;2 c;^{iB) C7„(a;) dz = m 9^ n
j Vr^{U^ix)}^dx = I
ORTHOGONAL SERIES
fix) = A^Uoix) + A^U.ix) + A^U^ix) +
A^ - - f' Vl^^fix) U^ix)
dx
RELATIONSHIPS BETWEEN TJx) AND C^J*!
n(x) = U^{x) - xU„^i{x)
{l-x^)U^.,{x) = xT^ix) - T^^Ax)
1 r' r^+iMdij
t^™{a:)
_ 1 p ^r.
a:)Vl--y2
r„(x)
1 W VT-^2C7„_,(^)
= -J
dv
GENERAL SOLUTION OF CHEBYSHEV'S DIFFERENTIAL EQUATION
V =
AT„ix) + By/T^^^U n-iix) il n= 1,2,3, ...
A + B sin-i a
if n=
31.1
HYPERGEOMETRIC DIFFERENTIAL EQUATION
x(l-x)y" + {c-(a+b + l)x)y' - aby =
Wl
RGEOMETRIC FUNCTIONS
A solution of 31.1 is given by
310 mn h-.-^\ _ 1 , a.6 ^ a(a + l)&(6 + l) a(a+ l)(a + 2)6(6 + 1)(6 + 2) . ^
31.2 Fia,b,c,x) - l+-^x+ i.2.c{« + l) ^'+ 1.2-3. c(c+l)(c-F2) *' +
If a, 6, c are real, then the series converges for — 1 < a; < 1 provided that c — (a + 6) > —1.
31.3 F(-p,l;l;-a:) = (! + «)"
31.4 F{l,l;2;-x) = [ln(l + JB)]/a!
31.5 lim F{l,n;l;x/n) = e*
31.6 F(^,-^;^;sin2a;) = cosx
31.7 F(^,l;l;sin2a;) = secx
SPECIAL CASE!
31.8
31.9 F{^,l;l;-x^) = (tan-iic)/a:
31.10 F{l,p;p;x) = 1/(1 - x)
31.11 F(n + l,-n;l;il-x)/2) = P„(a;)
31.12 F(n. -7i; ^; (1 - a:)/2) = r„(x)
GENERAL SOLUTION OF THE HYPERGEOMETRIC EQUATION
If c, a — 6 and c — a — b are all nonintegers, the general solution valid for ]«! < 1 is
31.13 y = A F{a, b; c; x) + Bx^~<' F(a- c + 1, b - c + l;2- c; x)
31.14
31.15
31.16
31.17
MISCELLANEOUS PROPERTIES
F(o, b; c; 1) =
r(o) r(c - g - 6)
T(c - a) r(c - 6)
—Fia, &; c;x) = — F(a + 1, 6 + 1; c + 1; x)
dx c ^
F{a,b:c;x) = r(6)r(c-&) J" ^"'Ml -^)'-'''-'(l -ux)"" du
F(a, 6; c; k) = (1 - a)c-a-bF(c - a, c - 6; c; x)
160
DEFINrriON o
^B^ApfAC^ffW^^^oP^fff
32.1
^{Fim = J €-'tF{t)dt = /(8)
In general /(a) will exist for s > a where a is some constant. .C is called the Laplace traneform
operator.
DEFINITION OF THE INVERSE LAPLACE TRANSFORM OF f{s)
If ^{F(e)> = /(s), then we aay that F(i) = ^-^{/(s)} is the inverse Laplace transform of /(s).
Jl~^ is called the inverse Laplace transform operator.
COMPLEX INVERSION FORMULi
The inverse Laplace transform of /(s) can be found directly by methods of complex variable theory.
The result is
32^
F{t)
c + fT
where c is chosen so that all the singular points of /(«) lie to the left of the line Re {s} = c in the complex
s plane.
161
162
LAPLACE TRANSFORMS
tE OF GENERAL PROPERTIES OF LAPLACE TRANSFORM
/(«)
F<t)
32.3
afAB) + bU(8)
oF,(() + &F2(()
32.4
fis/a)
a F(at)
32.5
f{B-a)
e«tF(t)
32.6
e-'»m
(Fit -a) t>a
[0 t<a
32.7
Bf(8)-F(0)
F'it)
32.8
82/(«)-«F(0)-F'(0)
F"(t)
32.9
8»f{8) - s"-»F(0) ~ stt-2F'(0) - • • • - F<«~i> (0)
FCn)(()
32.10
/*(•)
-tF(t)
32.11
/"(8)
t^F{t)
32.12
/(n)(g)
(_l)ntn p(t)
32.13
*
f F(u)du
*^0
32.14
a*
J].../^FM.«.=j;<;_%, FM.„
32.15
m 9(8)
r F{u)G{t~u)du
•^0
LAPLACE TRANSFORMS
163
/(s)
Fit)
32.16
j m)du
Fit)
t
32.17
^ r^
Fit) = Fit + T)
1 _ e-sTj " r-iujau
32.18
8
32.19
-fa/8)
8
1 Jo(2-\/wt)F(M)(£M
32.20
prn/a/^)
pin r M-n/2J„(2\/ut)F(M)dM
•^0
32.21
f(s + 1/s)
s^ + 1
r'
1 Jai2\/uit-u))Fiu)du
•^0
32.22
2v^^o
F(t2)
32.23
/(Ins)
« In s
f" t^Fiu) ,
J, Tiu+l)^''
32.34
P(s)
Q(s)
p(8) = polynomial of degree less than n,
Q(8) = {8- ai)(8 - aa) ■ ■ • (S - an)
where ai, aa. • ■ • , "n ^^^ all distinct.
4 P(«fe) ,«.t
164
LAPLACE TRANSFORMS
IB or
TSPITTCE TRANSFORMS
m
F(i)
32.25
1
8
1
32.26
1
S2
£
32.27
^ n-1,2,3....
^ , 0! = 1
{n-1)!'
32.28
^ «>o
r(7i)
32.29
1
8 — O
e«
32.30
fn— 1 aa(
, 0' = 1
(n-l)!'
(8_„)n " l,-.3....
32.31
(n-1 gat
Tin)
(a — a)"
32.32
1
82 + a2
sinaf
a
32.33
8
cos at
82 + a2
32.34
1
e^t sin at
a
(8 - 6)2 + a2
32.35
8- &
e^"' cos at
(8 - 6)2 + a2
32.36
1
sinh at
a
82 -a2
32.37
8
cosh a£
82 - a2
32.38
1
«''* sinh at
a
(8 - 6)2 - a2
LAPLACE TRANSFORMS
165
m
F(i)
32.39
8- b
e"" cosh at
(8 - 6)2 - o2
32.40
1 . ,
gbt — ^t
b-a
(s-a)(s-6) "^''
32.41
« . J
6 — a
(8-a)is~b) ''^'^
32.42
1
sin at — at cos at
2a3
(s2 + a2)2
32.43
8
£ sin at
2a
(S2 + a2)2
32.44
82
sin at + a( cos at
2a
(s2 + a2)2
32.45
83
cos o( — ^at sin at
(82 + a2)2
32.46
s2-a2
(s2 + a2)2
t cos at
32.47
1
at cosh ai — sinh at
2a3
(S2 - o2)2
32.48
«
£ sinh at
2a
(a2 - a2)2
32.49
«2
sinh a( + at cosh a(
2a
(82 _ a2)2
32.50
83
cosh at + -ia( sinh at '
(s2-a2)2
32.51
82+ a2
(82 - a2)a
t cosh ai
32.52
1
(3 - aH^) sin at - Sat cos at
8a5
(«2 + (i2)3
32.53
8
( sin at — at^ cos at
8a3
(82 + o2)«
32.54
S2
(1 + a2(2) sin at — at cos at
(82 + o2)8
8a3
32.55
a3
3t sin at + at2cosot
8a
(a2 + a2)3
166
LAPLACE TRANSFORMS
m
f(*)
32.56
s*
(3 — a^t^) sin ai + Qat cos o(
8a
(s2 + a2)3
32.57
S5
(8 — a^t^) cos a* — lat sin at
8
(s2 + a2)3
32.58
3s2 - o2
(s2 + a2)3
(2 sin at
2a
32.59
83 - 3a^s
(s2 + a2)3
^(2 cos at
32.60
(g2 + a2)4
^t^ cos at
32.61
g3 _ fl,2g
t3 sin at
24a
(g2 + 0,2)4
32.62
1
(3 + a2t2) sinh at — Sat cosh at
8a&
(«2 - a2)3
32.63
8
at2 cosh at ~ t sinh at
8a3
(s2 - a2)3
32.64
S2
at cosh at + (a2t2— 1) sinh at
8a3
(g2 _ (i2)3
32.65
S3
3f sinh at + at^ cosh at
8a
(82 - a2)3
32.66
81
(3 + a2t2) sinh at + 5at cosh at
8a
(82 _ 02)3
32.67
85
(8 + a2t2) cosh at + 7at sinh at
8
(8= - o2)3
32.68
3s2 + a2
(s2 - a2)3
t2 sinh at
2a
32.69
83 4- 3a28
(82 - a2)3
^(2 cosh at
32.70
«4 + 6a2s2 + a-*
(s2 - a2)4
■Jt3 cosh at
32.71
s3 + a^s
(82 - a2)4
t3 sinh at
24a
32.72
1
gat/z f /- . VSat v/Sat „ ,J
l-^|V3s.n^2 cos ^2 + e-3.t/2|
s3 + a3
LAPLACE TRANSFORMS
167
/(s)
F{t)
32.73
s
eat/2 J VSat , r- . VSat .,J
3a V°^ 2 + ^ ''" 2 ' 1
s3 + a3
32.74
S2
^3(.- + i:c"t/acos^/*)
s3 + a^
32.75
1
-:r{e3»--cos^,--V3.ni-}
8^ — a^
32.76
s
i_^{V3.n4--cos4-.e3..}
gS — (j3
32.77
82
1 / \/3 o A
g ( e«t + 2«-''t/2 cos ^^-^—j
s3 — a3
32.78
1
s4 + 4a4
^-g (sin at cosh at — cos a* sinh at)
32.79
s
sin a£ sinh at
2a2
s4 + 4o4
32.80
S2
^ (sin at cosh at + cos at sinh at)
«4 + 4a*
32.81
s3
cos at cosh at
s* + 4a4
32.82
1
^-g (sinh at — sin at)
s*-a4
32.83
8
. ■^-2(cosh at — cos at)
s4-a4
32.84
S2
g- (sinh at + sin at)
s4-a4
32.85
S3
■|-(cosh ot + cos at)
s4- a*
32.86
1
g-bt _ g-at
Vs + o + V* + 6
2(6 - o) V^«^
32.87
1
erf V^
sVs + a
32.88
1
e"* erf Vot
\/a
Vs (a — a)
32.89
1
gat j J^ 6 e**** erfc (6V^) I
y/8 — a+ b
168
LAPLACE TRANSFORMS
f(s)
i^(0
32.90
1
Jo (at)
Va^ + a2
32.91
1
/o(at)
Vs2 - a2
32.92
a"/„(aO
71 > —1
Vs2 + 0.2
32.93
a^IJat)
\/s2 - a2
32.94
gb(»- Vs^ + o')
Joia^/t{t + 2b))
■\/s2 + 02
32.95
g-W»»+a»
iJoia^/t^-b^) t>b
\o t<b
Vs2 + a2
32.96
1
tJiiat)
a
(s2 + a2)S/2
32.97
s
tJoiat)
(S2 + a2)3/2
32.98
82
J(i(at) — at J I (at)
(S2 + o2)3/2
32.99
1
tJiiat)
a
(S2 - a2)3.'2
32.100
s
tloiat)
(S2 - a2)3/2
32.101
S2
/o(at) + atl^iat)
(S2 ~ a2)3/2
32.102
1 e-*
F{t) = n, n ^ t < n + 1, n = 0, 1, 2, . . .
8{e»-l) s(l-e-»)
See also entry 32.165.
32.103
1 e-s
[ti
F{t) = ^r^
k = l
where \t] = greatest integer ^ (
8{es — r) 8(1 -re-")
32.104
8(e» — r) 8(1— re-")
See also entry 32.167.
F(t) = r", n^t<n+l, m = 0, 1, 2, . . .
32.105
g-a/s
cos 2^/at
LAPLACE TRANSFORMS
169
/(«)
Fit)
32.106
g-Q/S
B\Ti2y/ai
yfva
s3/2
32.107
e-o/s
/ \n/2
gn + l
32.108
e-aVs
V7t
Vb
32.109
e-aVa
o- •
2\/n-*3
32.110
s
erf ia/2-/i )
32.111
g-av's
erfc (a/2V^)
8
32.112
fi-av7
eb(bt+a) erfc f b\fi + -^\
\ 2VV
Vs (Vs + 6)
32.113
g-a/l^
gn+l
32.114
>»(m)
g-bt _ e-at
«
32.115
In [(s2 + a2)/a2]
28
Ci(ae)
32.116
In [{b + a)/a.]
8
Si(at)
32.117
(Y + In 8]
s
y = Euler's constant = .5772156...
Inf
32.118
in(s2 + b2y
2 (cos at — cos bt)
t
32.119
^2 (y + lns)2
68 a
y = Euler's constant = .5772156...
In^t
32.120
In 8
8
- (In e + 7)
y = Euler's constant = .5772156...
32.121
InZg
8
(In t + y}2 - ^JT^
y = Euler's constant = .5772156...
170
LAPLACE TRANSFORMS
m
F(0
32.122
r'(n + 1) - r(m + 1) In «
t" Int
32.123
tan~^ (a/s)
sin at
32.124
tan~' (a/s)
s
St (at)
32.125
gO/S .
erfc {yah )
g-2V^t
v^
32.126
/''*''' erfc (s/2o)
2a „v
32.127
/"*"' erfc (s/2a)
erf (at)
32.128
e"* erfc ^fas
1
Vir(f + a)
32.129
e*" Ei(as)
1
t + a
32.130
1
a
cos 08 -J^— Si(as) l — sin OS Ci{a^)
1
t2 + a2
32.131
ainoB <-— Si (as) y + cos as Ci (a«)
t
t2+ a2
32.132
cos as -J ^ — St (as) ^ — sin oa Ci {ax)
tan -3 {t/a)
8
32.133
sin as ■< ^ — Si (as) > 4- cos as Ci (as)
1, /t2 + aA
2'\ a2 ;
s
32.134
r "1^
1 , /t2 + a2\
-t'\ a2 ;
32.135
'^(0 = null function
32.136
1
8(t) = delta function
32.137
e-as
S(f-a)
32.138
S
See also entry 32.163.
Vit - a)
LAPLACE TRANSFORMS
171
/(»)
^(«)
32.139
sinh sa:
8 sinh sa
a; . 2 ^ (—1)" . mrx nirt
— 1- - i -^ — '— sin cos
a IT „ = 1 n a a
32.140
sinh 8x
8 cosh sa
4 ^ (-1)« . (2n-l)7ra; . (2n-lVt
- 2 o — H-sin^ r— ^ — sin^ — '- —
n- „= J 2n— 1 2a 2a
32.141
cosh sx
8 sinh as
( , 2 S (-1)" nvx . nirt
— 1 — > cos sin
a B- n"^! n a a
32.142
cosh sx
8 cosh sa
. , 4 ^ (-1)" (2n-lVa! (2n-lVt
1 H 2 TT -^ COS ^^ r — COS ^ ■=
T „ = i 2n— 1 2a 2a
32.143
sinh sx
8^ sinh sa
at , 2a ^ (—1)" . rtirx . nwt
1 5-2 ^ — 5^sin sin
a TT^ „=i n'' a a
32.144
sinh sx
, 8a ^ (-1)" . (2n-lVx (2n-lVt
X ^ ^ 2t 7S 7T9 Sin ^ cos x- — —
57^ „=i {2n — 1)2 2a 2a
s2 cosh sa
32.145
cosh 8X
8^ sinh sa
e2 , 2a ^ (-1)" nirx ( , niTi\
„ + 9 2 9 cos 1 cos 1
2a TT^ „=( n2 a \ a /
32.146
cosh sx
s^ cosh sa
. , 8a ^ (-1)" (2m -IM . <2n - IM
^+^2^(2UV°^ 2a «•" 2a
32.147
cosh sa;
s3 cosh sa
W + x2 a2) ^^"^^ i <-l^" cos^^^^-^^'^^os^^""^''^*
32.148
sinh xVT
sinh aVT
^ 2 ( l)««e--*'^''-'8in'^"-='
a n=l a
32.149
cosh x-\f8
cosh oV?
h 2 {-l)«-M2n-l)e-<2n-i>Vt/4a'cos^^^^^^^
a' „=i ^a
32.150
sinh x\fB
yfs cosh aVi
- 2 (-l)n-l«-(2«-l>Vt/4a'sin^^^^^^
a „-i ^a
32.151
cosh x^f8
yfa sinh aVs
1+2 1 (_ij„^-nVWe08^^
a a „=i a
32.152
sinhxVfi^
s sinh aVs
a a- „= 1 n a
32.153
cosh a:Vfi
8 cosh aVs
1 + ^2 <-^'>-c2«-nVt/4««,o,(2r^-lV«
a- ^■r'j 2?i — 1 2a
32.154
sinh xVs^
s2 sinh oVs
xt , 2a2 " (-1)",, „*^t/„»v ■ «'^a;
32.155
cosh a:V*
s2 cosh aVs
4(X2 a^\ + t 1^"' -5 ^-^*" «-(2n-l)Vt/4a« cos'^"'^^"''
172
LAPLACE TRANSFORMS
fie)
F{t)
32.156
JgiixVs)
8 Jq (iay/T)
where X^, X2» ■ ■ • ^'^ *^® positive roots of Jq{\) =
32.157
n=l Xn-ZitM
where X^, \2i • ■ ■ are the positive roots of ^o(^) ~ ^
Triangular wave function
F{t)
32.158
a«2*^"^(2)
1-
/\/\/\/
2a 4a 6a
Fig. 32-1
Square wave function
32.159
itanh(f)
1-
F(t)
la i 2a 1 3a i 4a ' 5a
1 I ! I L^
Fig. 32-2
Rectified sine wave function
Fm
32.160
e2«2 + ^coth(^2J
1-
^\/^\/^\/
a 2a 3a '
Fig. 32-3
Half rectified sine wave function
Fit)
32.161
vd
1-
r\ r\ /
(a282 + ^)(l-e-af)
a 2a Sa 4a
Fig. 32-4
Saw tooth wave function
FW
32.162
1 e-<^
1-
^^\^^\^
aa^ 8(1-6-08)
a 2a 3a 4a
Fig. 32-5
LAPLACE TRANSFORMS
173
32.163
32.164
32.165
32.166
32.167
32.168
f(s)
s
See also entry 32.138.
e-os (! — «-")
s{l - 6-"^)
See also entry 32.102.
e(l - e-*)''
1- e-
8(1 — re-«)
See also entry 32.104.
a2s2 + 7r2
Fit)
Heavislde's unit function V{t — a)
F{t)
1-
Flg. 32-6
Pulse function
Fit)
1-
Fig. 32-7
Step function
3-
2-
1
2a
— I—
3a
Fig. 32-8
Fit) =n2, 71 S t<n+l, 71 = 0, 1,2,
Fig. 32-9
F(() = r«, n ^ ( < n + 1, n = 0, 1, 2,
2 3
Fig. 32-10
Fit) =
sin {Trt/a) ^ t g a
t > a
Fig. 32-11
— I—
4a
F(t)
1
1
3-
2-
1-
1 \ ^ ^
T '
33.1
where
33.2
FOI
^m
W-iSR^
f{^) = I MW cos aX + B{a) sin ax) da
If"
A (a) — ~ I fix) COS ax dx
1 C"
B{a) = — I f{x) sin ax dx
Sufficient conditions under which this theorem holds are:
(i) f{x) and f'(x) are piecewise continuous in every finite interval —L < x < L;
(ii) 1 |/(a;)| dx converges;
(iii) f(x) is replaced by ^{f{x + 0) + f{x — 0)} if a; is a point of discontinuity.
33.3
33.4
EQUIVALENT FORMS OF FOURIER'S INTEGRAL THEOREM
fix
) = 2~ J j ^^^^ *^°^ aix-u)
■^rth^; .A *'*! — M^
du da
a= —30 "11= —
fix) = ^ C e'«* da ( fill) e-*«" du
du da
— 00 — 1
33.5 /(a;) = — I sin ax da I /(w) sin au du
where fix) is an odd function {f{—x) = —/(a:)].
33.6 /(a;) = — ( cos aa; da | /(m) cos au du
where fix) is an even /uncfcion [f(~x) =f(x)].
174
FOURIER TRANSFORMS
175
The Fourier transform of f(x) is defined as
33.7 ?"{/(«)> = ^(«) = J /(a;)e-*«*dx
Then from 33.7 the inverse Fourier transform of F(a) is
33.8 f-HFia)) = f{^) = ^J ^(«)e*«*d«
We call f{x) and F{a) Fourier transform pairs.
CONVOLUTION THEOREM FOR FOURIER TRANSFORMS
If F(a) = Tifix)) and G(a) = Tiffi^)}, then
33.9 ^ f F{a) G{a) e^=^ da = J fiu)g(x-u)
where f*g is called the convolution of / and g. Thus
33.10 7{f*ff} = nnriff}
du = f*g
EVAL'S IDENTITY
If Fia) = nfix)}, then
33.11 J" \fix)\^dx = ^y \F{a)\^da
More generally if F(a) = f{f(x)} and G{a) = ^{^(a!)}, then
33.12 C fix)^)dx = ^ r F{a)GW)da
where the bar denotes complex conjugate.
FOURIER SINE TRANSFORMS
The Fourier sine transform of f{x) is defined as
33.13 Fsia) = Tsifix)) = r f{x)amaxdx
Then from 33.13 the inverse Fourier sine transform of Fsia) is
2 r°°
33.14 fix) = frM-PsW} = -\ Fsia)Binaxda
176
FOURIER TRANSFORMS
dx
The Fourier cosine transform of fix) is defined as
33.15 Fc(a) = TciH^c)) = f f{x)cosax
Then from 33.15 the inverse Fourier cosine transform of Fc(a) is
33.16 fix) = f-HFcM) = Ij Fc(«)cosaxd«
TRANSFORM PAI
fix)
J^(«)
33.17
jl \x\ < b
\o \x\ > b
2 sin 6a
a
33.18
1
b
X^+ 62
33.19
X
--g-bot
o
X^+ 62
33.20
/(n)(a:)
t"a"F(a)
33.21
x^fix)
33.22
fibx)e^
iK-r)
FOURIER TRANSFORMS
177
SPECIAl FOURIiR SINE TRANSFORMS
33.23
33.24
33.25
33.26
33.27
33.28
33.29
33.30
33.31
33.32
33.33
33.34
33.35
33.36
/(x)
1 < a: < 6
a; > 6
»-»
a:2 + 62
,-bi
a;"—) g—bx
,-bI*
--1/2
a;-"
Bin bx
Bin bx
cos bx
tan-i (rc/b)
csc&z
e2x_i
fcM
1 — cos 6a
He-ba
2
a2+ 62
r(n) sin (n tan-W6)
(a2 + 62)n/2
463/2 "«
/4b
v^
ya"~l CSC (ntr/2)
2r(7i)
< m < 2
1 , fa + b
2^n;r^
B^a/2 a < 6
3r6/2 a> b
' a < 6
7r/4 a=b
r-/2 a > 6
2^*"''"
26**"*^ 26
f-^K?)~^
178
FOURIER TRANSFORMS
SPECIAL FOURIER COSINE TRANSFORMS
/(*)
Fcia)
33.37
Jl < X < &
to a; > 6
sin 6a
a
33.38
1
26
a2 + (,2
33.39
g~bx
b
a2 + 62
33.40
a;n-l 0-bx
r{»i) cos (n tan^i a/6)
(«2 + 62)n/2
33.41
e-bx'
2\6
33.42
a: -1/2
v^
33.43
a;-n
^«'-isec(W2) o<«<l
2 r(n)
33.44
g-ca _ e-*><^
■sra
33.45
sin bx
X
rw/2 a<b
• 7r/4 a = 6
[ a > 6
33.46
sin bx^
\ IT ( cfi . a^\
V86V=°^46-^^"46;
33.47
cos bx^
V^('°'i^+'^"I^)
33.48
sech bx
^^^^i
33.49
cosh (V^ x/2)
cosh iV^x)
n? cosh iVTra/2)
\ 2 cosh (Vw- a)
33.50
e-b^
■^{cos(26V^) - sin(26v^)>
V^
ELLIPTIC FUNCTIONS
INCOMPLETE ELirPTIC rNTEGRAL OF THE FIRST KIND
34.1
u = F{k
'*' = r
d9
s:
dv
where ^ = am u is called the amplitude of u and x = sin 0, and where here and below < fe < 1.
COMPLETE ELLIPTIC INTEGRAL OF THE FIRST KIND
34.2 K = F(ft,W2) = (
de
Vl - *^ sjn2 9
-i:
dv
= fii + (lT*= + (i^T'^ + (i^"^ +
a
INCOMPLETE ELLIPTIC INTEGRAL OF THE SECOND KIND
34.3
Vl - A:2 sin2 tf d9 = I ^
n ''o Vl -
Vl-u2
ffPW
THE^SECOND KT
Vl - fc2 sin2 ^ d# =: I
^0
■1 Vl - fc2^2
Vl-tj2
dv
= ii-(iT--feyf-(i^TT
INCOMPLETE ELLIPTIC INTEGRAL OF THE THIRD KIND
34.5
de
= /:
dv
(1 + n sin2 «) Vl - ^^ sin2 « ^^ (1 +mv2) V(l -■u2)(l -ft2i;2)
179
180
ELLIPTIC FUNCTIONS
34.6
COMPUTE ELLIPTIC INTEGRAL OF THE THIRD KIND
da _ r^ dv
•J n
Xir/2
(1 + m 8in2 «) vT-fc^sin^ ^o (1 + nv^) y/(\ - v^){l - k^v^)
LANDEN'S TRANSFORMATION
34.7
This yields
34.8
sin 201
*^"^ = fc + co8 20, *''■ ''^'"'^ " sm[2^,-4>)
F(fc.0) = f
d9
Vl - fc^ 8in2 tf
2 f*'
1 + feJ.
dffi
\/l-feJsin2(?i
where k^ — 2\Ic/{l + k). By successive applications, sequences kj, feg. ^3. • - - and <f>i, ^g, 03, . . . are obtained
such that ft < fci < fcg < feg < ■ ■ - < 1 where lim fe„ = 1. It follows that
34.9
where
34.10
. 2\/fe , 2V^ ^ ^
The result is used in the approximate evaluation of F(ft,0).
JACOBI'S ELLIPTIC FUNCTIONS
From 34.1 we deiine the following elliptic functions.
34. 11 X = sin (am w) = sn m
34.12 v1^^k2 — cos (am m) = cn«
34. 1 3 Vl - *=^*^ = Vl -ft2sn2M = dn w
"We can also define the inverse functions sn-' x, cn~^ x, dn-i x and the following
34.14 nsM ~
34.15 nc« —
34.16 ndw =
sn u
1
cnw
1
dnw
34.17
sc w =
34.18 sd« - ^
dnu
34.19 cd« = ^
dnu
34.20
CSM —
snu
34.21
. dnM
dcM =
cnw
34.22
dnu
dsM — — — ■
snM
34.23
34.24
34.25
sn (m + v)
en (m + v)
dn (m + V)
SUM cni* dntJ + cnM sni) dna
1 — A;2 sn2 u 8n2 v
COM en V — snasnudnwdnu
1 - A;2 sn2 M sn2 11
dntt dnt? — ft^ snu sni; cna cn-u
1 — fc2 an2 y gn2 ^
ELLIPTIC FUNCTIONS
181
34.26 ^sna - cnudnM
34.27 ^ en u - - sn u dn M
34.28 -2-dn«
du
34,29 ^scu
— — fc2 sn M en u
— dc u nc M
34.30
34.31
34.32
SERIES EXPANSIONS
cnu = i-|^+(i + 4fc2) j^ - {1 + 4Ak2 + 16k*) fy + • ■ •
2 A
dnw = 1 - A:2|y+ fe2(4 + fc2)^_ fc2(i6 + 44fc2 + A^)^+ ...
34.33
lf'K,k = If f" "'"^
12 32 ^ 52
= .915965594.
Let
34.34
Then
34.35
34.36
34.37
PERIODS OF ELLli>TIC FUNCTIONS
K =
^0 Vl - fc2 gin2 ' J,
ff/2
\/l-fe'2sin2ff
snM has periods 4^ and 2iii:'
en u has periods 4K and Sif + 2iK'
dnu has periods 2K and 4iK'
where A:' = \1~!^
IDENTtTIES INVOLVING EUIPTIC FUNCTIONS
34.38 sn2tt + cn^w = 1
34.40 dn2u-A:2cn2« - fe" where )fc' - a/T^^
34.42
34.44
CH'^M =
dn 2m + en 2m
VI
1 + dn 2u
— en 2m sn m dn m
+ cn2u
cnu
34.39 dn2u + fc2sn2u = 1
1 - en 2m
34.41
sn-'u —
1 + dn 2m
34.43 dn2„ = 1 - fcg + dn 2m + fe' en w
1 + dn 2m
34.45
Vf
— dn 2u ft sn u en u
+ dn2M
dnu
182
ELLIPTIC FUNCTIONS
mmmm
34.46 snO = 34.47 cnO = 1 34.48 dn = 1 34.49 scO = 34.50 am =
INTEGRALS
34.51 I snuiiu = -r In (dn m — /c en u)
34.52 j en u du = -j- cos~i (dn u)
34.53 I dnwdw = sin~^snM)
34.54 I sc u du
■ , In (dc u + y/T—l^ nc «)
34.55 I cs u du = In (ns m — ds m)
34.56 \ cdudu
-r In {nd M + fc ad u)
34.57 I dc u du
34.58 fsdudu
In (nc M + ac m)
-1
In (na m — ca «)
sin~^ (fe edit)
34.59 JdaudM
34.60 I ns M dw = In (ds u — cs «)
34.61 I ncu
34.62 I nd u du
du =
1
1 , /, , sew
In dc M +
VH^
coB~i (cdu)
LEGENDRE'S RELATION
34.63
where
34.64
34.65
EK' + E'K - KK' = v/2
E
- r^
fc2 8in2 e d$
•^ n
■n/2
dg
J-'tt/Z , /•jr/2 fla
-^0 Vir^fc^i
Vr^^fc2sin2ff
sin^ff
ERROR FUNCTION erf (a;) = — r f e""
du
35.1 erf(x) = ~(x-^+^2, ^.3,
+
35.2 erf («)
1 -
1 -7^4 +
1-3 1*3'5
+
^^\- 2a:2 ' (2a;2)2 (2x2)3
35.3 erf(-x) = -erf(x), erf (0) ^ 0, erf('») = 1
COMPLEMENTARY ERROR FUNCTION erfc(a:) = 1 - erf (x) = — = f e "' d«
35.4 erfc («)
35.5 erfc (a;)
35.6 erfc (0)
2_ / _ a:3 gs _ z?
V^r 3-1! ^5-2! 7-3!
+
yfi^xV 23:2 ' (2a:2)2 (2x2)S
1, erfc{<") =
J"" fi-"
— -
du
35.7 Ei{x) =
35.8 £i(a^) =
35.9 Eiix) -
^i 1
—7 — In X + I -
— y — \nx +
_ »— u
dw
+
I'l! 2-2! ' 3*3!
Ill 1-11+2|_|| +
35.10 EH-^) =
35.11 Si(x) =
SINE INTEGRAL
x3 _^ rS
Si{x) = f
({U
1-1! 3-3! 5*5! 7'7!
£ _ sing /l _ 3! 5! _
2 a; la; arS a:5
+
35.12 Si{x) ~
35.13 Si(-x) = -Si(a:), Si(0) = 0, Si(«) = W2
cos a; / _ 2! 4! _
X \^ ^"^^
183
184 MISCELLANEOUS SPECIAL FUNCTIONS
COSINE INTEGRAL Ci{x) = C'^^^du
35.14 Ci{x) = -y-lnx+ Ci^l^^^du
J,
35.15 Ci(x) = -y - \nx + ^ - ^^ + ^ ^ + ■-■
2-2! 4-4! 6-6! 8-8!
35.16 Ci(x) ~ ^^^(l-^+^ A-Sin^/ 2!,4! \
X \x x^^ x^ J X y- x'^^x* J
35.17 Ci('*) =
FRESNEL SINE INTEGRAL S{x) = J^ j'Jsinu^ du
\^V3-1! 7-3! 11-5! 15-7! ^
35.19 Six) ~ l-~^Jico.x2)(l~llA+ll3^_ __\ /j^_l^S^
2 ^/2^\ V 22a;5 2^x^ J ^ ^"^ " ' ' \2x^ 23ar'
35.20 S(-x) = -Six), 5(0) = 0, S{«) = -^
FRESNEL COSINE INTEGRAL C{x) = .pfcostt^dtt
35.2, C(.) =^(^-_^,_4_^_-.,...)
35
35.23 a-x) - -C(a;), C(0) - 0, C(~) - 1
RIEMANN ZETA FUNCTION ^a:) = TF + 4 + 4 +
35.24 «., = ^r"-Hilf_,,, ,>!
35.25 ni-a;) = 2i-^;r-^r(a:) co3(o-a:/2)j:(a:) [extension to other values]
-Iy2fc
22k-V2fc5
35.26 m) = ~7^J~ k = 1,2.3....
TRIANGLE INEQUALITY
36.1
36.2
|ai] - Ittal ^ lai + ojl ^ [aj + [oal
|ai + fla + ■ . ■ + a„| g ]a,I + [ojl + . - . + |a„|
CAUCHY-SCHWAR2 INEQUALITY
36.3 \a,b, + 0362 + ■ . . + a„feJ2 ^ (|a,|2 + la^I^ + • - ■ + \aM\h\^ + I62P + ' • " + i6„|2)
The equality holds if and only if ai/6, - ag/fca = ■ • • = a„/6„.
INEQUALITIES INVOLVING ARITHMETIC, GEOMETRIC AND HARMONIC MEANS
If ^ . G and H are the arithmetic, geometric and harmonic means of the positive numbers a,, a,,..., a,
H ^ G ^ A
then
36.4
where
36.5 A = "i + °2 + • • • + '^n
36.6 G = ^a,a^...a, 36.7 ^ .= 1 f 1 + 1 + . . . + A
The equality holds if and only if aj = Og = • ■ • = a„.
H - nVo +a + •■• +
36.8
where
36.9
HOLDER'S INEQUALITY
kifri + ^262 + • ■ ■ + a„6„| ^ (|a,|p + la^lP + . . . + Kji>)1/p(|6,|<. + [ft^i^ + . . . + |(,j,ji/.
- + ^ = 1 P>1. 9>1
duces
The equality holds if and only if W-Vil-.l = Wp-./|6,| = • • • = |a,|.-V|M, For p = , = 2 it
re-
186
186
INEQUALITIES
CHEBYSHEV'S INEQUALITY
If tti ^ a2 ^ ' ■ ■ ^ a„ and 61 S 63 = ■ ■ • = &„> then
n
n
n
36.10
or
36.11 {ai + a2+ ■•■ +«„)(&! + &£+ ••■+&„) = n(ai&i + a262+ •■ ■ + a„6„)
MINKOWSKI'S INEQUALITY
If Qi, a^, . . ., a„, bi, 6,, . . . b„ are all positive and p > 1, then
36.12 {(ai+fe,)P+ (02+62)''+ ■■■ + (On+fiJ'*}^^'' ^ (ai + 02+ ■■■+0^^" + ib'l + bl+ ■■■ + 6^)i/p
The equality holds if and only if O1/61 = 02/62 = ■ ■ - = o.n^f>n-
CAUCHY-SCHWARZ INEQUALITY FOR INTEGRALS
I/:
36.13 II f{x)g{x)dx
The equality holds if and only if f{x)/g(x) is a constant.
j''\f{x)\-^dx\Ij^\g{x)\^dx
HOLDER'S INEQUALITY FOR INTEGRALS
36.14 J \fix)g{x)\dx ^ if \f{x)\''dxi \j \9{x)\''dx
where l/p + 1/g = 1, p > 1, g > 1. If p = g - 2, this reduces to 36.13.
The equality holds if and only if \f{x)\f-^/\g{x)\ is a constant.
i/n
MINKOWSKI'S INEQUALITY FOR INTEGRALS
If P > 1,
36.15
<j \nx) + g(x)\^dx^ ^ ij \nx)\^dx^ + U \9(x)\'>dx
i/p
The equality holds if and only if f{x)/g{x) is a constant.
37.1 cot a; =
X "^ 2a: j^2_^+^2_4^2 + x2_9^ +
37.2
CSCX - ^ ^^\^z_^ -2^2 + ^^—^
37.3 sec X = Av
' .+ =
.2 _ 4a;2 9^.2 _ 4a;2 ^ 2B;r2 - 4a:2
37.4 tan x = 8a; -I ^ h ^ I- . ^ u
lan ic »* \2 _ 4a;2 ^ 9^ - 43^2 + 26772 - 4a:2 ^
37.5
sec^a; =
4 J 1 + 1 + 1 , 1 ,
37.6 csc^ a; =
-1+ 1 + 1 + 1 I 1 +
a;2 (ar-;r)2 ^ (a: + ,r)2 ^ (a: - 2,r)2 ^ (a; + 2^)2 ^
37.7 coth X =
1 + 0-.}^L—^ 1 , 1 .
X U2 + ,r2 ^ ^ + 4^ "^ a:2 + 9^ +
37.8 csch X =
l_2x-^-^^ ^
X |a;2 + B^ a;2 + 4^ "f' ^2 + 9^
1 + ^
37.9 sechx = 4^
^+ 5
572 + 4x2 9^2 + 43.2 ^ 25^ + 43.2
37.10 tanhx =
^* "^ s-2 -f 4x2 "•■ 9^ + 4a;2 + 25:r2 + 4x2 "^
187
3B.1
sina: =
.u_-Vi_- ,_^
38.2
C08X =
-^Vi-|S)(i-^.
38.3 sinh x
^(i + 5)(i + 5)(i+i.
38.4 cosh X =
-^)(-is)(^-^;
See also 16.12, page 102.
38.6 MX) = 1-^ 1-- 1-
where Xj, X2, X3, . . . are the positive roots of Jq{x) = 0.
38.7 JM = .fi-gyi-E!yi_E^
where \i, X2, X3> . . . are the positive roots of Ji(x) = 0.
38.8
sin X _
X
X X X X
COS - cos — COS -- cos — -
2 4 8 16
38.9
2 2 4 4 6 6
1 ' 3* 3' 5'5'7
This is called Wallis' product.
188
BINOMIAL DISTRIBUTION
39.1
p>0, g>0, p + q = 1
POISSON DISTRIBUTION
39.2
*(«:)= 2 ^ X >
HYPERGEOMETRIC DISTRIBUTION
39.3
*(«) = 2
t/\n — t
t<x /r + s
n
NORMAL DISTRIBUTION
39.4
X) = ^f e-'Ut
<i.(x) =
STUDENT'S t DISTRIBUTION
39.5
y/^ ~r(n/2)
1 V 2 / /•■
*^^* = JZZ T(n/2) J
1+^
n
-(n + I)/2
d(
CHI SQUARE DISTRIBUTION
39.6
1 r^
F DISTRIBUTION
39.7
*(«) =
r(n,/2) r(n2/2)
r t"./2(n2 + nit)-<"i+»*5/2d(
189
40
SPECIAL MOMENTS
OF INERTIA
The table below shows the moments of inertia of various rigfid bodies of mass M. In all cases it is
assumed the body has uniform [i.e. constant] density.
TYPE OF RIGID BODY
MOMENT OF INERTIA
40.1 Thin rod of length a
(a) about axis perpendicular to the rod through the center of
mass,
(6) about axis perpendicular to the rod through one end.
40.2 Rectangular parallelepiped with sides a, b, c
J^M(a2 + (,2)
(a) about axis parallel to c and through center of face ah,
{V) about axis through center of face be and parallel to c.
40.3 Thin rectangular plate with sides a,b
JLM(a2 + 62)
(a) about axis perpendicular to the plate through center,
(6) about axis parallel to side 6 through center.
40.4 Circular cylinder of radius a and height h
(a) about axis of cylinder,
(6) about axis through center of mass and perpendicular to
cylindrical axis,
(c) about axis coinciding with diameter at one end.
._ - Hollow circular cylinder of outer radius a,
inner radius 6 and height h
^Af {a2 + 62)
JLM(3a2 + 362 + fe2)
r^MiZa^ + 362 + 4^2)
(a) about axis of cylinder,
(6) about axis through center of mass and perpendicular to
cylindrical axis,
(c) about axis coinciding with diameter at one end.
190
SPECIAL MOMENTS OF INERTIA
191
40.6 Circular plate of radius a
iMa2
(a) about axis perpendicular to plate through center,
(6) about axis coinciding with a diameter.
-_ _ Hollow circular plate or ring with outer radius a
and inner radius 6
^M{a^ + 62)
lM(a2 + 62)
(a) about axis perpendicular to plane of plate through center,
(6) about axis coinciding with a diameter.
40.8 Thin circular ring of radius a
Ma2
^Ma2
(a) about axis perpendicular to plane of ring through center,
(6) about axis coinciding with diameter.
40.9 Sphere of radius a
§Mo2
lMa2
(a) about axis coinciding with a diameter,
(6) about axis tangent to the surface.
40.10 Hollow sphere of outer radius a and inner radius 6
f M(a5 - 65)/((i3 - 68)
f M(a5 - 65)/(a3 - 63) + Ma^
(a) about axis coinciding with a diameter,
(6) about axis tangent to the surface.
40.11 Hollow spherical shell of radius a
2Ma2
(a) about axis coinciding with a diameter,
(6) about axis tangent to the surface.
40.12 Ellipsoid with semi-axes a,b,c
|M(6a2 + 62)
(a) about axis coinciding with semi-axia c,
(6) about axis tangent to surface, parallel to semi-axis c and
at distance a from center.
40. 1 3 Circular cone of radius a and height h
^M(a2 + 4fe2)
|M(4a2 + fe2)
(a) about axis of cone,
(6) about axis through vertex and perpendicular to axis,
(c) about axis through center of mass and perpendicular to axis.
40.14 Torus with outer radius a and inner radius 6
lM{Qa^ - lOab + 562)
(a) about axis through center of mass and perpendicular to
plane of torus,
(6) about axis through center of mass and in the plane of the
torus.
Length
Area
Volume
Speed
Density
Force
Energy
Power
Pressure
1 kilometer (km)
1 meter (m)
1 centimeter (cm)
1 millimeter (mm)
1 micron (ju)
1 millimicron (m^u)
1 angstrom (A)
— 1000 meters (m)
— 100 centimeters (cm)
= 10-2 m
^ 10-3 m
— 10-8 m
= 10-!' m
= 10-10 m
1 inch (in.)
1 foot (ft)
1 mile (mi)
1 mil
1 centimeter
1 meter
1 kilometer
2.540 cm
30.48 cm
1.609 km
10-3 in.
0.3937 in.
39.37 in.
0.6214 mile
1 square meter (m2) ^ 10.76 ft^
1 square foot (f t2) - 929 cm2
1 square mile (mi2) = 640 acres
1 acre = 43,560 ft2
1 liter (/) = 1000 cm^ = 1.057 quart (qt) - 61.02 in3 = 0.03532 ft^
1 cubic meter (m^) = 1000 I = 35.32 ft^
1 cubic foot (ft3) - 7.481 U.S. gal = 0.02832 m^ - 28.32 I
1 U.S. gallon (gal) ^ 231 in^ - 3.785 I; 1 British gallon = 1.201 U.S. gallon = 277.4 in*
1 kilogram (kg) =^ 2.2046 pounds (lb) = 0.06852 slug; lib = 453.6 gm = 0.03108 slug
1 slug = 32.174 lb = 14.59 kg
1 km/hr ^ 0.2778 m/sec ^ 0.6214 rai/hr = 0.9113 ft/sec
1 mi/hr = 1.467 ft/sec = 1.609 km/hr = 0.4470 m/sec
1 gm/cm3 = 103 kg/m3 = 62.43 Ib/ft^ = 1.940 slug/ft3
1 lb/ft3 = 0.01602 gm/cm3; 1 slug/ft^ - 0.5154 gm/cm^
1 newton (nt) - 10^ dynes = 0.1020 kgwt = 0.2248 Ibwt
1 pound weight (Ibwt) ~ 4.448 nt = 0.4536 kgwt = 32.17 poundais
1 kilogram weight (kgwt) ^ 2.205 Ibwt = 9.807 nt
1 U.S. short ton = 2000 Ibwt; 1 long ton = 2240 Ibwt; 1 metric ton = 2205 Ibwt
1 joule = Intm = 10^ ergs = 0.7376 ft Ibwt ^ 0.2389 cal = 9.481 X lO^^ Btu
1 ft Ibwt ^ 1.356 joules ^ 0.3239 cal = 1.285 X 10-3 Btu
1 calorie (cal) = 4.186 joules = 3.087 ft Ibwt = 3.968 X 10-3 Btu
1 Btu (British thermal unit) - 778 ft Ibwt = 1055 joules = 0.293 watt hr
1 kilowatt hour (kw hr) = 3.60 X 10« joules - 860.0 kcal = 3413 Btu
1 electron volt (ev) = 1.602 X IQ-^^ joule
1 watt = 1 joule/sec = 10' ergs/sec = 0.2389 cal/sec
1 horsepower (hp) ^ 550 ft Ibwt/sec = 33,000 ft Ibwt/min = 745.7 watts
1 kilowatt (kw) = 1.341 hp = 737.6 ft Ibwt/sec = 0.9483 Btu/sec
1 nt/m^ - 10 dynes/cm2 = 9.869 X 10-8 atmosphere = 2.089 X 10-2 lbwt/ft2
1 lbwt/in2 - 6895 nt/m^ = 5.171 cm mercury = 27.68 in. water
1 atmosphere (atm) = 1.013 X 10^ nt/m2 = 1.013 X 10^ dyne3/cm2 = 14.70 lbwt/in2
= 76 cm mercury = 406.8 in. water
192
Part n
TABLES
SAMPLE PROBLEMS
ILLUSTRATING USE OF THE TABLES
COMMON LOGARITHMS
1. Find log 2.36.
We must find the number p such that lO" - 2.36 = A'. Since 10" = 1 and 10^ = 10, p lies
between and 1 and can be found from the tables of common logarithms on page 202.
Thus to find log 2.36 we glance down the left column headed N until we come to the first two digits,
23. Then we proceed right to the column headed 6. We find the entry 3729. Thus log 2.36 = 0.3729,
ie. 2.36 = 100-3'29.
2. Find (a) log 23.6, (6) log 236, (c) log 2360.
From Problem 1, 2.36 = 10''-3729, jhen multiplying successively by 10 we have
23.6 = 10'-3'29, 236 = 102-3729^ 2360 = 103-3729
Thus
(a) log 23.6 = 1.3729
(b) log 236 = 2.3729
(c) log 2360 = 3.3729.
The number .3729 obtained from the table is called the mantissa of the logarithm. The number
before the decimal point is called the characteristic. Thus in (6) the characteristic is 2.
The following rule is easily demonstrated.
Rule 1. For a number greater than 1, the characteristic is one less than the number of digits before
the decimal point. For example since 2360 has four digits before the decimal point, the
characteristic is 4 — 1 — 3.
3. Find (a) log .236, (6) log .0236, (c) log .00236.
From Problem 1, 2.36 = 10''-3'='29. Then dividing successively by 10 we have
.236 = 10*>3ra9-J = 1Q9.3729-10 = 10~-8271
.0236 = 100-3729-2 = X08-3729-10 == 10-16271
.00236 = 1003729-3 - IO'3729-IO = 10-2-6271
Then
(a) log .236 ^ 9.3729 - 10 - -.6271
(6) log .0236 ^ 8.3729 - 10 ^ -1.6271
(c) log .00236 ^ 7.3729 - 10 =^ -2.6271.
The number .3729 is the mantissa of the logarithm. The number apart from the mantissa [for
example 9 - 10, 8 — 10 or 7 — 10] is the characteristic.
The following rule is easily demonstrated.
Rule 2. For a positive number less than 1, the characteristic is negative and numerically one more
than the number of zeros immediately following the decimal point. For example since .00236
has two zeros immediately following the decimal point, the characteristic is —3 or 7 — 10.
194
SAMPLE PROBLEMS ILLUSTRATING USE OF THE TABLES 195
4. Verify each of the following logarithms.
(a) log87.2. Mantissa - .9405. characteristic - 1; then log 87.2 = 1.9405.
(b) log 395,000 = 5.5966.
(c) log .0482. Mantissa - .6830, characteristic - 8 - 10; then log.o .0482 - 8.6830 - 10.
(rf) log .000827 - 6.9175 - 10.
5. Find log 4.638.
Mantissa of log 4640 .. .6665 Mantissa of log 4.638 = .6656 + (.8)(.0009)
Mantissa of log 4630 = .6656 _ (.RRt * * j- -^
— .d663 to four digits
Tabular difference - .0009 Then log 4.638 = 0.6663
(6656+')!"'' '"' P^°P-*--^ P-ts table on page 202 can be used to give the mantissa directly
6. Verify each of the following logarithms.
(a) log 183.2 ^ 2.2630 (2625 + 5)
(6) log 87,640 - 4.9427 (9425 + 2)
(c) log .2548 =9.4062-10 (4048 + 14)
(d) log .009848 = 7.933 - 10 (9930 + 3)
COMMON ANTILOGARITHMS
7. Find (a) antilog 1.7530, (b) antilog (7.7530 - 10)
digits before the decimal point. T.^TeZ^f, .ll^Z'i^ '' '' ^'^^^ ^'^ '^^
^'* ac'tlris'ril'V-'lo^th?' '"h' 5662 corresponding to the mantissa .7530. Then since the char-
Thnrfl I' ""^^"^ """'* ^"^^^ **° ^«^°s immediately following the decimal noint
Ihus the required number is .005662. ^"'luwing me aecimal point.
8. Find antilog (9.3842 - 10).
on page%rwlTa;f' ' "" '^'"^^" ''''' ^"' •'''' »"^ ^ ™-* "^ interpolation. From the table
Number corresponding to.3850 = 2427 Given mantissa = 3842
Number corresponding to .3840 = ^42]_ Next smaller mantissa - .3840
Tabular difference = 6 Difference = "i^^
Then 2421 + ^(2427-2421) ^ 2422 to four digits, and the required number is 0.2422.
The proportional parts table on page 204 can also be used.
9. Verify each of the following antilogarithm.
(a) antilog 2.6715 = 469.3
(b) antilog 9.6089 -10 = .4063
(c) antilog 4.2023 = 15.930
196 SAMPLE PROBLEMS ILLUSTRATING USE OF THE TABLES
COMPUTATIONS USING LOGARITHMS
j^ ^ ^ (784.6)(.0431) ^ j^gp ^ log 784.6+ log .0431 - log 28.23.
28.23
log 784.6 = 2.8947
(+) log .0431 =^ 8.6345-10
11.5292-10
{-) log 28.23 = 1.4507
logP = 10.0785-10 = .0785. Then P = 1.198.
Note the exponential significance of the computation, i.e.
(784.6)(.0431) ^ (102.B347)(10a.e345-10) ^ ^q2.,M7 + S.^Z,B-10-IA,07 = 10.0785 ^ 1198
28.23 101-4507
11. P = (5.395;8. logP = 8 log 5.395 = 8(0.7320) - 5.8560, and P ^ 717,800.
12. P = V38^2 = (387.2)>/2. logP = | log 387.2 = ^(2.5879) = 1.2940 and P = 19.68.
13. P = ^.08317 = (.08317)1/5. log P - ^ log .08317 = ^(8.9200-10) = f, (48.9200 - 50) - 9.7840-10
and P = .6081.
14. P = V-003654 ( 18-3 7P j^^p ^ 4 log .003654 + 3 log 18.37 - (4 log 8.724 + ^ log 743.8)
(8.724)4 ^743.8
Numerator N Denominator D
I log .003654 - ^(7.5628-10) 4 log 8.724 = 4(0.9407) = 3.7628
= ^(17.5628-20) = 8.7814-10 ^ log 743.6 = 1(2.8714) = 0.7178
3 log 18.37 ^ 3(1.2641) = 3.7923 Add: log D = 4.4806
Add: logiV = 12.5737-10
logN = 12.5737-10
(-) logZ) - 4.4806
logP =^ 8.0931-10. Then P = .01239
NATURAL OR NAPIERIAN LOGARITHMS
15. Find (a) In 7.236, (6) In 836.2, (c) In .002548.
(a) Use the table on page 225.
In 7.240 = 1.97962
In 7.230 = 1.97824
Tabular diflference = .00138
Then In 7.236 ^ 1.97824 4- ^(.00138) = 1.97907
In terms of exponentials this means that e^-^'^o' - 7.236.
(6) As in part (a) we find
In 8.362 = 2.12346 + 3^^(2.12465 - 2.12346) = 2.12370
Then
In 836.2 ^ In (8.362 X 102) = log 8.362 + 2 In 10 ^ 2.12370 + 4.60517 = 6.72887
In terms of exponentials this means that ee-'^ss? = 836.2.
(c) As in part (a) we find
In 2.548 - 0.93216 + 3%{0.93609 - 0.93216) = 0.93530
In .002548 = In (2.548 X lO"-*) = In 2.548 - 3 In 10 = 0.93530-6.90776 - -5.97246
In terms of exponentials this means that g- 5-97246 - .002548.
SAMPLE PROBLEMS ILLUSTRATING USE OF THE TABLES X97
TRIGONOMETRIC FUNCTIONS (DEGREES AND MINUTES)
16. Find (a) sin 74°23', (6) cos 35°42', (c) tan 82°56'.
(a) Refer to the table on page 206.
sin 74°30' = .9636
sin 74°20' = .9628
Tabular difference - .0008
'^*'*" sin 74<'23' - .9628 + ^(.0008) = .9630
{&) Refer to the table on page 207.
cos35''40' - .8124
cos 35^60' = .8107
Tabular difference — .0017
"T^^^" cos 35^42' - .8124 - ^(.0017) - .8121
" cos 35^42' ^ .8107 + ^(.0017) = .8121
(c) Refer to the table on page 208.
tan82°60' = tan 83°0' = 8.1443
tan82°50' ^ 7.9530
Tabular difference = .1913
■^^^^ tan82°56' = 7.9530 + -^^(.1913) = 8.0678
17. Find (a)cot45°16', (6) sec 73°48', (c) esc 28°33'.
(a) Refer to the table on page 209.
cot45°10' = .9942
cot45°20' := .9884
Tabular difference = .0058
^*^^" cot45°16' = .9942 - ^(.0058) = .9907
°' cot45'^I6' - .9884 + ^(.0058) = .9907
(6) Refer to the table on page 210.
sec73°50' - 3.592
sec73°40' = 3.556
Tabular difference =: .036
'^''^" sec 73^48' = 3.556 + ^(.036) = 3.585
(c) Refer to the table on page 211.
csc28'=30' - 2.096
csc28''40' = 2.085
Tabular difference = ,011
■"•^ csc28='33' - 2.096 - ^(.011) = 2.093
°' csc28°33' = 2.085+ ^^(.011) - 2.093
198 SAMPLE PROBLEMS ILLUSTRATING USE OF THE TABLES
INVERSE TRIGONOMETRIC FUNCTIONS (DEGREES AND MINUTES)
18. Find (a) sin-i (-2143), (6) cos-> (.5412), (c) tan-i (1.1536).
(a) Refer to the table on page 206.
sin 12^30' = .2164
sinl2°20' = .2136
Tabular difference = .0028
Since .2143 is -^l-^^ ~ -2136 _ j ^^ ^^^ ^ between .2136 and .2164, the required angle is
.0028 *
12'*20' + i(lO') = 12°22.5'.
(6) Refer to the table on page 207.
cos57°10' ^ .5422
cos 57=^20' = .5398
Tabular difference - .0024
Then cos-M-5412) = 57°20' - i^^^^^^^^dO') - 57^4.2'
or eos->(.5412) - 57n0' + i^l?^=2^(10'} - 57^4.2'
(e) Refer to the table on page 208.
tan49°10' = 1.1571
tan49°0' ^ 1.1504
Tabular difference — .0067
Then tan-M1.1536) = 4900- + 1-1536^- 1.1504 ^^p,^ ^ ^904.8'
Other inverse trigonometric functions can be obtained similarly.
TRIGONOMETRIC AND INVERSE TRIGONOMETRIC FUNCTIONS (RADIANS)
19. Find (a) sin (.627), (6) cos (1.056), (c) tan (.153).
(a) Refer to the table on page 213.
sin (.630) - .58914
sin (.620) = .58104
Tabular difference — .00810
Then sin (.627) - .58104 + -^^(.00810) = .58671
(6) Refer to the table on page 214.
cos (1.050) = .49757
cos (1.060) = .48887
Tabular difference = .00870
Then cos (1.056) = .49757 - ^(.00870) = .49235
or cos (1.056) = .48887 + ^(.00870) = .49235
(c) Refer to the table on page 212.
tan (.160) = .16138
tan (.150) = .15114
Tabular difference — .01024
Then tan (.153) = .15114 + ^(.01024) = .15421
Similarly other trigonometric functions are obtained.
SAMPLE PROBLEMS ILLUSTRATING USE OF THE TABLES 199
20. Find sin"' (.512j in radians.
Refer to the table on page 213.
sin (.540) ~ .51414
sin (.530) = .50553
Tabular difference = .00861
.512 - .50553
.00861
Similarly the other inverse trigonometric functions are obtained.
Then sin-M.512) - .530 + iii^--J?^ f.Ol) - .5375 radians
COMMON LOGARITHMS OF TRIGONOMETRIC FUNCTIONS
21. Find (a) log sin 63 = 17', (6) log cos 48=" 44'.
(a) Refer to the table on page 217.
Iogsin63°20' = 9.9512 - XO
logsin63°10' = 9.9505-10
Tabular difference = .0007
Then log sin 63°17' = 9.9505 - 10 -f -j:^(.0007) = 9.9510 - 10
(6) Refer to the table on page 219.
log cos 48°40' = 9.8198 - 10
logcos48°50' = 9.8184-10
Tabular difference = .0014
Then logcos48°44' = 9.8198 - 10 - 3^(.0014) = 9.8192-10
o^ logcos48°44' ^ 9.8184 - 10 + -j^(.0014) = 9.8192-10
Similarly we can find logarithms of other trigonometric functions. Note that log sec x = -log cos x,
log cot ic — -log tan a;, log esc ac — -log sin x.
22. If log tan a: - 9.6845-10, finds:.
Refer to the table on page 220.
Iogtan25°50' = 9.6850-10
]ogtan25°40' = 9.6817-10
Tabular difference = .0033
Then :. = 25°40- + ^-^«^5- 9.6817 ^^Q,^ ^ ^^.^^ g,
.UUoo
CONVERSION OF DEGREES, MINUTES AND SECONDS TO RADIANS
23. Find 75° 28' 47" in radians.
Refer to the table on page 223.
70° = 1.221730 radians
5° = .087267
20' - .005818
8' = .002327
40" = .000194
7^ - .000034
Adding, 75° 28' 47" = 1.317370 radians
200 SAMPLE PROBLEMS ILLUSTRATING USE OF THE TABLES
CONVERSION OF RADIANS TO DEGREES, MINUTES AND SECONDS
24. Find 2.5-17 radians in degrees, minutes and seconds.
Refer to the table on page 222.
2 radians - 114° 35' 29.6"
.6 = 28° 38' 52.4"
.04 = 2° 17' 30.6"
.007 = 0° 24' 3.9"
Adding, 2.547 radians = 144° 114' 116.5" - 145° 55' 56.5"
CONVERSION OF RADIANS TO FRACTIONS OF A DEGREE
25. Find 1.382 radians in terms of degrees.
Refer to the table on page 222.
1 radian = 57.2958°
.3 = 17.1887°
.08 = 4.5837°
.002 = .1146°
Adding, 1.382 radians = 79.1828°
EXPONENTIAL AND HYPERBOLIC FUNCTIONS
2G. Find (a) e->-^'*, (fc) e--i5B.
(a) Refer to the table on page 226.
e5.30 ^ 200.34
e5.2o ^ 181.27
Tabular difference = 19.07
Then «5.24 = I81.27 + -^(19.01) = 188.90
(6) Refer to the table on page 227.
e--i50 = ,86071
e-.i6o ^ .85214
Tabular difference = .00857
Then e" >S8 = .86071 - ^^(.00857) = .86385
or e->58 = .85214 + ^(.00857) = .85385
27. Find (ct) sinh (4.846), (b) sech (.163).
(a) Refer to the table on page 229.
sinh (4.850) = 63.866
sinh (4.840) = 63.231
Tabular difference ~ .635
Then sinh (4.846) = 4.840 + ^(.635) = 5.221
(6) Refer to the table on page 230.
cosh (.170) = 1.0145
cosh (.160) = 1.0128
Tabular diflference = .0017
Then cosh (.163) = 1.0128 + ^(.0017) = 1.0133
and so sech (.163) = = — - — = .98687
^ cosh (.163) 1.0133
SAMPLE PROBLEMS ILLUSTRATING USE OF THE TABLES 201
28. Find tanh-' (.71423).
Refer to the table on page 232.
tanh (.900) = .71630
tanh(.890) - .71139
Tabular difference = .00491
Then tanh-' (.71423) = .890 + ^HM^^L^Zll^dO) = .8958
.00491 '
INTEREST AND ANNUITIES
29. A man deposits $2800 in a bank which pays 5% compounded quarterly. What will the deposit
amount to in 8 years?
There are n = 8 • 4 = 32 payment periods at interest rate r - .05/4 - .0125 per period. Then the
amount is
A = $2800(1 + .0125)32 = $2800(1.4881) = $4166.68
using the table on page 240.
30. A man expects to receive $12,000 in 10 years. How much is that money worth now, considering interest
at 6% compounded semi-annually?
We^are asked for the present value P which will amount to A - $12,000 in 10 years. Since there
are k - 10 • 2 ^ 20 payment periods at interest rate r = .06/2 = .03 per period, the present value is
P = $12,000(1 + .03)- 2" = $12,000(.55368) = $6644.16
using the table on page 241.
31. An investor has an annuity in which a payment of $500 is made at the end of each year. If interest
is 4% compounded annually, what is the amount of the annuity after 20 years?
Here r = .04, n = 20 and the amount is [see table on page 242],
$500
" (1 + .04)20 _ I
.04
$500(29.7781) = $14,889.05
32. What is the present value of an annuity of $120 at the end of each 3 months for 12 years at 6%
compounded quarterly?
Here n = 4 • 12 = 48 payment periods, r = .06/4 =: .015 and the present value is
$120
.015
using the table on page 243.
L .0
- (1.015)
= $120(34.0426) = $4085.11
TABLE
1
FOUR PLACE COMMON LOGARITHMS
logioA^ or log A^
N
1
2
3
4
5
6
7
8
9
Proportion
al Parts
I 2
3
4
5
6
7
8 9
10
0000
0043
0086
0128
0170
0212
0253
0294
0334
0374
4 8
12
17
21
25
29
33 37
n
0414
0453
0492
0531
0569
0607
0645
0682
0719
0755
4 8
11
15
19
23
26
30 34
12
0792
0828
0864
0899
0934
0969
1004
1038
1072
1106
3 7
10
14
17
21
24
28 31
13
1139
1173
1206
1239
1271
1303
1335
1367
1399
1430
3 6
10
13
16
19
23
26 29
14
1461
1492
1523
1553
1584
1614
1644
1673
1703
1732
3 6
9
12
15
18
21
24 27
15
1761
1790
1818
1847
1875
1903
1931
1959
1987
2014
3 6
8
n
14
17
20
22 25
16
2041
2068
2095
2122
2148
2175
2201
2227
2253
2279
3 5
8
11
13
16
18
21 24
17
2304
2330
2355
2380
2405
2430
2455
2480
2504
2529
2 5
7
10
12
15
17
20 22
18
2553
2577
2601
2625
2648
2672
2695
2718
2742
2765
2 5
7
9
12
14
16
19 21
19
2788
2810
2833
2856
2878
2900
2923
2945
2967
2989
2 4
7
9
U
13
16
18 20
20
3010
3032
3054
3075
3096
3118
3139
3160
3181
3201
2 4
6
8
11
13
15
17 19
21
3222
3243
3263
3284
3304
3324
3345
3365
3386
3404
2 4
6
8
10
12
14
16 18
22
3424
3444
3464
3483
3502
3522
3541
3560
3579
3598
2 4
6
8
10
12
14
15 17
23
3617
3636
3655
3674
3692
3711
3729
3747
3766
3784
2 4
6
7
9
11
13
15 17
24
3802
3820
3838
3856
3874
3892
3909
3927
3945
3962
2 4
5
7
9
11
12
14 16
25
3979
3997
4014
4031
4048
4065
4082
4099
4116
4133
2 3
5
7
9
10
12
14 15
26
4150
4166
4183
4200
4216
4232
4249
4265
4281
4298
2 3
5
7
8
10
11
13 15
27
4314
4330
4346
4362
4378
4393
4409
4425
4440
4456
2 3
5
6
8
9
11
13 14
28
4472
4487
4502
4518
4533
4548
4564
4579
4594
4609
2 3
5
6
8
9
11
12 14
29
4624
4639
4654
4669
4683
4698
4713
4728
4742
4757
1 3
4
6
7
9
10
12 13
30
4771
4786
4800
4814
4829
4843
4857
4871
4886
4900
1 3
4
6
7
9
10
11 13
31
4914
4928
4942
4955
4969
4983
4997
5011
5024
5038
1 3
4
6
7
8
10
11 12
32
5051
5065
5079
5092
5105
6119
5132
5145
5159
5172
1 3
4
5
7
8
9
11 12
33
5185
5198
6211
5224
5237
5250
5263
5276
5289
5302
1 3
4
6
6
8
9
10 12
34
5315
5328
5340
5353
5366
5378
5391
5403
5416
5428
1 3
4
5
6
8
9
10 11
35
6441
5453
5465
5478
5490
5502
5514
5527
5539
5551
1 2
4
5
6
7
9
10 U
36
5563
5575
5587
5599
5611
5623
5635
5647
6658
5670
1 2
4
5
6
7
8
10 11
37
5682
5694
5705
5717
5729
5740
5752
5763
6775
5786
1 2
3
5
6
7
8
9 10
38
5798
5809
5821
5832
5843
5855
5866
5877
5888
5899
1 2
3
5
6
7
8
9 10
39
5911
5922
5933
5944
5955
5966
5977
5988
5999
6010
1 2
3
4
5
7
8
9 10
40
6021
6031
6042
6053
6064
6075
6085
6096
6107
6117
1 2
3
4
5
6
8
9 10
41
6128
6138
6149
6160
6170
6180
6191
6201
6212
6222
1 2
3
4
5
6
7
8 9
42
6232
6243
6253
6263
6274
6284
6294
6304
6314
6325
1 2
3
4
5
6
7
8 9
43
6335
6345
6355
6365
6375
6385
6395
6406
6415
6425
1 2
3
4
5
6
7
8 9
44
6435
6444
6464
6464
6474
6484
6493
6503
6513
6522
1 2
3
4
5
6
7
8 9
45
6632
6542
6551
6561
6571
6580
6590
6599
6609
6618
1 2
3
4
5
6
7
8 9
46
6628
6637
6646
6656
6665
6675
6684
6693
6702
6712
1 2
3
4
5
6
7
7 8
47
6721
6730
6739
6749
6758
6767
6776
6786
6794
6803
1 2
3
4
5
5
6
7 8
48
6812
6821
6830
6839
6848
6857
6866
6875
6884
6893
1 2
3
4
4
5
6
7 8
49
6902
6911
6920
6928
6937
6946
6955
6964
6972
6981
1 2
3
4
4
5
6
7 8
50
6990
6998
7007
7016
7024
7033
7042
7050
7059
7067
1 2
3
3
4
5
6
7 8
51
7076
7084
7093
7101
7110
7118
7126
7135
7143
7152
1 2
3
3
4
5
6
7 8
52
7160
7168
7177
7185
7193
7202
7210
7218
7226
7235
1 2
2
3
4
5
6
7 7
53
7243
7251
7259
7267
7275
7284
7292
7300
7308
7316
1 2
2
3
4
6
6
6 7
54
7324
7332
7340
7348
7356
7364
7372
7380
7388
7396
1 2
2
3
4
5
6
6 7
N
1
2
3
4
5
6
7
8
9
1 2
3
4
5
6
7
8 9
202
Tabre 1
FOUR PLACE COMMON LOGARITHMS
(continued)
logio A^
or I09
A^
N
1 2
3
4
5 6
7
8
9
Proportional Parts
1
2
3
4
5
6
7
8
9
55
7404
7412 7419
7427
7435
7443 7451
7459
7466
7474
2
2
3
4
5
5
6
7
56
7482
7490 7497
7505
7513
7520 7528
7536
7643
7551
2
2
3
4
5
5
6
7
57
7559
7566 7574
7582
7589
7597 7604
7612
7619
7627
2
2
3
4
5
5
6
7
58
7634
7642 7649
7657
7664
7672 7679
7686
7694
7701
2
3
4
4
5
6
7
59
7709
7716 7723
7731
7738
7745 7752
7760
7767
7774
2
3
4
4
5
6
7
60
7782
7789 7796
7803
7810
7818 7825
7832
7839
7846
2
3
4
4
5
6
6
61
7853
7860 7868
7875
7882
7889 7896
7903
7910
7917
2
3
4
4
5
6
6
62
7924
7931 7938
7945
7952
7959 7966
7973
7980
7987
2
3
3
4
5
6
6
63
7993
8000 8007
8014
8021
8028 8035
8041
8048
8055
2
3
3
4
5
5
6
64
8062
8069 8075
8082
8089
8096 8102
8109
8116
8122
2
3
3
4
5
5
6
65
8129
8136 8142
8149
8156
8162 8169
8176
8182
8189
2
3
3
4
5
5
6
66
8195
8202 8209
8215
8222
8228 8235
8241
8248
8254
2
3
3
4
5
5
6
67
8261
8267 8274
8280
8287
8293 8299
8306
8312
8319
2
3
3
4
5
5
6
68
8325
8331 8338
8344
8351
8357 8363
8370
8376
8382
2
3
3
4
4
5
6
69
8388
8395 8401
8407
8414
8420 8426
8432
8439
8445
2
2
3
4
4
5
6
70
8451
8457 8463
8470
8476
8482 8488
8494
8500
8506
2
2
3
4
4
5
6
71
8513
8519 8525
8531
8537
8543 8549
8555
8561
8567
2
2
3
4
4
5
5
72
8573
8579 8585
8591
8597
8603 8609
8615
8621
8627
2
2
3
4
4
5
5
73
8633
8639 8645
8651
8657
8663 8669
8675
8681
8686
2
2
3
4
4
5
5
74
8692
8698 8704
8710
8716
8722 8727
8733
8739
8745
2
2
3
4
4
5
5
75
8751
8756 8762
8768
8774
8779 8785
8791
8797
8802
2
2
3
3
4
5
5
76
8808
8814 8820
8825
8831
8837 8842
8848
8854
8859
2
2
3
3
4
5
5
77
8865
8871 8876
8882
8887
8893 8899
8904
8910
8915
2
2
3
3
4
4
5
78
8921
8927 8932
8938
8943
8949 8954
8960
8965
8971
2
2
3
3
4
4
5
79
8976
8982 8987
8993
8998
9004 9009
9015
9020
9025
2
2
3
3
4
4
5
80
9031
9036 9042
9047
9053
9058 9063
9069
9074
9079
2
2
3
3
4
4
5
81
9085
9090 9096
9101
9106
9112 9117
9122
9128
9133
2
2
3
3
4
4
5
82
9138
9143 9149
9154
9159
9165 9170
9175
9180
9186
2
2
3
3
4
4
5
83
9191
9196 9201
9206
9212
9217 9222
9227
9232
9238
2
2
3
3
4
4
5
84
9243
9248 9253
9258
9263
9269 9274
9279
9284
9289
2
2
3
3
4
4
5
85
9294
9299 9304
9309
9315
9320 9325
9330
9335
9340
2
2
3
3
4
4
5
86
9345
9350 9355
9360
9365
9370 9375
9380
9385
9390
2
2
3
3
4
4
5
87
9395
9400 9405
9410
9415
9420 9425
9430
9435
9440
2
2
3
3
4
4
88
9445
9450 9455
9460
9465
9469 9474
9479
9484
9489
2
2
3
3
4
4
89
9494
9499 9504
9509
9513
9518 9523
9528
9533
9538
2
2
3
3
4
4
90
9542
9547 9552
9557
9562
9566 9571
9576
9581
9686
2
2
3
3
4
4
91
9590
9595 9600
9605
9609
9614 9619
9624
9628
9633
2
2
3
3
4
4
92
9638
9643 9647
9652
9657
9661 9666
9671
9675
9680
2
2
3
3
4
4
93
9685
9689 9694
9699
9703
9708 9713
9717
9722
9727
2
2
3
3
4
4
94
9731
9736 9741
9745
9750
9754 9759
9763
9768
9773
2
2
3
3
4
4
95
9777
9782 9786
9791
9795
9800 9805
9809
9814
9818
2
2
3
3
4
4
96
9823
9827 9832
9836
9841
9845 9850
9854
9859
9863
2
2
3
3
4
4
97
9868
9872 9877
9881
9886
9890 9894
9899
9903
9908
2
2
3
3
4
4
98
9912
9917 9921
9926
9930
9934 9939
9943
9948
9952
2
2
3
3
4
4
99
9956
9961 9965
9969
9974
9978 9983
9987
9991
9996
2
2
3
3
3
4
N
1 2
3
4
5 6
7
8
9
1
2
3
4
5
6
7
8
9
203
TABLE
2
FOUR PLACE COMMON ANTILOGARITHMS
10^ or antilog p
p
1
2
3
4
5
6
7
8
9
Proportional Parts
1
2
3
4
5
6
7
8 9
.00
1000
1002
1005
1007
1009
1012
1014
1016
1019
1021
1
2
2 2
.01
1023
1026
1028
1030
1033
1035
1038
1040
1042
1045
1
2
2 2
.02
1047
1050
1052
1054
1057
1059
1062
1064
1067
1069
1
2
2 2
.03
1072
1074
1076
1079
1081
1084
1086
1089
1091
1094
1
2
2 2
.04
1096
1099
1102
1104
1107
1109
1112
1114
1117
1119
2
2
2 2
.05
1122
1125
1127
1130
1132
1135
1138
1140
1143
1146
2
2
2 2
.06
1148
1151
1153
1156
1159
1161
1164
1X67
1169
1172
2
2
2 2
.07
1175
1178
1180
1183
1186
1189
1191
1194
1197
1199
2
2
2 2
.08
1202
1205
1208
1211
1213
1216
1219
1222
1225
1227
2
2
2 3
.09
1230
1233
1236
1239
1242
1245
1247
1250
1253
1256
2
2
2 3
.10
1259
1262
1265
1268
1271
1274
1276
1279
1282
1285
2
2
2 3
.11
1288
1291
1294
1297
1300
1303
1306
1309
1312
1315
2
2
2
2 3
.12
1318
1321
1324
1327
1330
1334
1337
1340
1343
1346
2
2
2
2 3
.13
1349
1352
1355
1358
1361
1365
1368
1371
1374
1377
2
2
2
3 3
.14
1380
1384
1387
1390
1393
1396
1400
1403
1406
1409
2
2
2
3 3
.15
1413
1416
1419
1422
1426
1429
1432
1435
1439
1442
2
2
2
3 3
.16
1445
1449
1452
1455
1459
1462
1466
1469
1472
1476
2
2
2
3 3
.17
1479
1483
1486
1489
1493
1496
1500
1503
1507
1510
2
2
2
3 3
.18
1514
1517
1521
1524
1528
1531
1535
1538
1542
1545
2
2
2
3 3
.19
1549
1552
1556
1560
1563
1567
1570
1574
1578
1581
2
2
3
3 3
.20
1585
1589
1592
1596
1600
1603
1607
1611
1614
1618
2
2
3
3 3
.21
1622
1626
1629
1633
1637
1641
1644
1648
1652
1656
2
2
2
3
3 3
.22
1660
1663
1667
1671
1675
1679
1683
1687
1690
1694
2
2
2
3
3 3
.23
1698
1702
1706
1710
1714
1718
1722
1726
1730
1734
2
2
2
3
3 4
.24
1738
1742
1746
1750
1754
1758
1762
1766
1770
1774
2
2
2
3
3 4
.25
1778
1782
1786
1791
1795
1799
1803
1807
1811
1816
2
2
2
3
3 4
.26
1820
1824
1828
1832
1837
1841
1845
1849
1854
1858
2
2
3
3
3 4
.27
1862
1866
1871
1875
1879
1884
1888
1892
1897
1901
2
2
3
3
3 4
.28
1905
1910
1914
1919
1923
1928
1932
1936
1941
1945
2
2
3
3
4 4
.29
1950
1954
1959
1963
1968
1972
1977
1982
1986
1991
2
2
3
3
4 4
.30
1995
2000
2004
2009
2014
2018
2023
2028
2032
2037
2
2
3
3
4 4
.31
2042
2046
2051
2056
2061
2065
2070
2075
2080
2084
2
2
3
3
4 4
.32
2089
2094
2099
2104
2109
2113
2118
2123
2128
2133
2
2
3
3
4 4
.33
2138
2143
2148
2153
2158
2163
2168
2173
2178
2183
2
2
3
3
4 4
.34
2188
2193
2198
2203
2208
2213
2218
2223
2228
2234
1
2
2
3
3
4
4 5
.35
2239
2244
2249
2254
2259
2265
2270
2275
2280
2286
1
2
2
3
3
4
4 5
.36
2291
2296
2301
2307
2312
2317
2323
2328
2333
2339
1
2
2
3
3
4
4 5
.37
2344
2350
2355
2360
2366
2371
2377
2382
2388
2393
1
2
2
3
3
4
4 5
.38
2399
2404
2410
2415
2421
2427
2432
2438
2443
2449
1
2
2
3
3
4
4 5
.39
2455
2460
2466
2472
2477
2483
2489
2495
2500
2506
1
2
2
3
3
4
5 5
.40
2512
2518
2523
2529
2535
2541
2547
2553
2559
2564
1
2
2
3
4
4
5 5
.41
2570
2576
2582
2588
2594
2600
2606
2612
2618
2624
1
2
2
3
4
4
5 5
.42
2630
2636
2642
2649
2655
2661
2667
2673
2679
2685
1
2
2
3
4
4
5 6
.43
2692
2698
2704
2710
2716
2723
2729
2735
2742
2748
1
2
3
3
4
4
5 6
.44
2754
2761
2767
2773
2780
2786
2793
2799
2805
2812
1
2
3
3
4
4
5 6
.45
2818
2825
2831
2838
2844
2851
2858
2864
2871
2877
1
2
3
3
4
5
5 6
.46
2884
2891
2897
2904
2911
2917
2924
2931
2938
2944
1
2
3
3
4
5
5 6
.47
2951
2958
2965
2972
2979
2985
2992
2999
3006
3013
1
2
3
3
4
5
5 6
.48
3020
3027
3034
3041
3048
3055
3062
3069
3076
3083
1
2
3
4
4
5
6 6
.49
3090
3097
3105
3112
3119
3126
3133
3141
3148
3155
1
2
3
4
4
5
6 6
P
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8 9
204
Table 2
(continued)
FOUR PLACE COMMON ANTILOG ARITHMS
10^ or antilog p
p
.50
.51
.52
.53
.54
.55
.56
.57
.58
.59
.60
.61
.62
.63
.64
.65
.66
.67
.68
.69
.70
.71
.72
.73
.74
.75
.76
.77
.78
.79
.80
.81
.82
.83
.84
.85
.86
.87
.88
.89
.90
.91
.92
.93
.94
.95
.96
.97
.98
.99
3162 3170 3177 3184 3192
3236 3243 3251 3258 3266
3311 3319 3327 3334 3342
3388 3396 3404 3412 3420
3467 3475 3483 3491 3499
3548 3556 3565 3573 3581
3631 3639 3648 3656 3664
3715 3724 3733 3741 3750
3802 3811 3819 3828 3837
8890 3899 3908 3917 3926
3981 3990 3999
4074 4083 4093
4169 4178 4188
4266 4276 4285
4365 4375 4385
4467 4477 4487
4571 4581 4592
4677 4688 4699
4786 4797 4808
4898 4909 4920
5012 5023 5035
5129 5140 5152
5248 5260 5272
5370 5383 5395
5495 5508 5521
5623 5636 5649
5754 5768 5781
5888 5902 5916
6026 6039 6053
6166 6180 6194
6310 6324 6339
6457 6471 6486
6607 6622 6037
6761 6776 6792
6918 6934 6950
7079 7096 7112
7244 7261 7278
7413 7430 7447
7586 7603 7621
7762 7780 7798
7943 7962 7980
8128 8147 8166
8318 8337 8356
8511 8531 8551
8710 8730 8750
8913 8933 8954
9120 9141 9162
9333 9354 9376
9550 9572 9594
9772 9795 9817
4009 4018
4102 4111
4198 4207
4295 4305
4395 4406
4498 4508
4603 4613
4710 4721
4819 4831
4932 4943
5047 5058
5164 5176
5284 5297
5408 5420
5534 5546
5662 5675
5794 5808
5929 5943
6067 6081
6209 6223
6353 6368
6501 6516
6653 6668
6808 6823
6966 6982
7129 7145
7295 7311
7464 7482
7638 7656
7816 7834
7998 8017
8185 8204
8375 8395
8570 8590
8770 8790
8974 8995
9183 9204
9397 9419
9616 9638
9840 9863
3199 3206 3214 3221 3228
3273 3281 3289 3296 3304
3350 3357 3365 3373 3381
3428 3436 3443 3451 3459
3508 3516 3524 3532 3540
3589 3597 3606 3614 3622
3673 3681 3690 3698 3707
3758 3767 3776 3784 3793
3846 3855 3864 3873 3882
3936 3945 3954 3963 3972
4027 4036 4046 4055 4064
4121 4130 4140 4150 4159
4217 4227 4236 4246 4256
4315 4325 4335 4345 4355
4416 4426 4436 4446 4457
4519 4529 4539 4550 4560
4624 4634 4645 4656 4667
4732 4742 4753 4764 4775
4842 4853 4864 4875 4887
4955 4966 4977 4989 5000
5070 5082
5188 5200
5309 5321
5433 5445
5559 5572
5689
5821
5957
6095
6237
6383
6531
6683
6839
6998
7161
7328
7499
7674
7852
8035
8222
8414
8610
8810
9016
9226
9441
9661
9886
5702
5834
5970
6109
6252
6397
6546
6699
6855
7015
7178
7345
7516
7691
7870
8054
8241
8433
8630
8831
9036
9247
9462
9683
9908
5093 5105 5117
5212 5224 5236
5333 5346 5358
5458 5470 5483
5585 5598 5610
5715 5728 5741
5848 5861 5875
5984 5998 6012
6124 6138 6152
6266 6281 6295
6412
6561
6714
6871
7031
7194
7362
7534
7709
7889
8072
8260
8453
8650
8851
9057
9268
9484
9705
9931
6427
6577
6730
6887
7047
7211
7379
7551
7727
7907
8091
8279
8472
8670
8872
9078
9290
9506
9727
9954
6442
6592
6745
6902
7063
7228
7396
7568
7745
7925
8110
8299
8492
8690
8892
9099
9311
9528
9750
9977
9
Proportional Parts
123456789
2 3 4 4 5 6
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
2 4
2 4
2 4
2 4
12 4
1 2 4
1 2 4
1 3 4
1 3 4
13 4
1 3 4
1 3 4
1 3 4
1 3 4
4
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
7
7
2 3 4 6 5 6 7
4 5 6 6
4 5 6 7
4 5 6 7
4 5 6 7
4 5 6 7
4 5
4 5
4 5
4 6
5 6
5 6
5 6
5 6
5 6
5 6
5 7
5 7
5 7
6 7
6 7
6 7 9 10 12 13
6 8 9 11 12 14
6 8 9 11 12 14
6 8 9 11 13 14
6 8 10 11 13 15
7 8 10 12 13 15
7 8 10 12 13 15
7 9 10 12 14 16
7 9 11 12 14 16
7 9 11 13 14 16
6
7
8
9
6
7
9
10
7
8
9
10
7
8
9
10
7
8
9
10
7
8
9
11
7
8
10
11
7
9
10
11
8
9
10
11
8
9
10
12
8
9
10
12
8
9
11
12
8
10
11
12
8
10
11
13
9
10
11
13
2 5 7
7 9 11
8 9 11
8 10 12
8 10 12
8 10 12
8 10 12
8 11 13
9 11 13
9 11 13
9 11 14
13 15 17
13 15 17
14 15 17
14 16 18
14 16 18
15 17 19
15 17 19
15 17 20
16 18 20
16 18 20
12 3 4 5
205
X
0'
10'
20'
30'
40'
50'
0°
.0000
.0029
.0058
.0087
.0116
.0145
1
.0175
.0204
.0233
.0262
.0291
.0320
2
.0349
.0378
.0407
.0436
.0465
.0494
3
.0523
.0552
.0581
.0610
.0640
.0669
4
.0698
.0727
.0756
.0785
.0814
.0843
5°
.0872
.0901
.0929
.0958
.0987
.1016
6
.1045
.1074
.1103
.1132
.1161
.1190
7
.1219
.1248
.1276
.1305
.1334
.1363
8
.1392
.1421
.1449
.1478
.1507
.1536
9
.1564
.1593
.1622
.1650
.1679
.1708
10°
.1736
.1765
.1794
.1822
.1851
.1880
11
.1908
.1937
.1965
.1994
.2022
.2051
12
.2079
.2108
.2136
.2164
.2193
.2221
13
.2250
.2278
.2306
.2334
.2363
.2391
14
.2419
.2447
.2476
.2504
.2532
.2560
15°
.2588
.2616
.2644
.2672
.2700
.2728
16
.2756
.2784
.2812
.2840
.2868
.2896
17
.2924
.2952
.2979
.3007
.3035
.3062
18
.3090
.3118
.3145
.3173
.3201
.3228
19
.3256
.3283
.3311
.3338
.3365
.3393
20°
.3420
.3448
.3475
.3502
.3529
.3557
21
.3584
.3611
.3638
.3665
.3692
.3719
22
.3746
.3773
.3800
.3827
.3854
.3881
23
.3907
.3934
.3961
.3987
.4014
.4041
24
.4067
.4094
.4120
.4147
.4173
.4200
25°
.4226
.4253
.4279
.4305
.4331
.4358
26
.4384
.4410
.4436
.4462
.4488
.4514
27
.4540
.4566
.4592
.4617
.4643
.4669
28
.4695
.4720
.4746
.4772
.4797
.4823
29
.4848
.4874
.4899
.4924
.4950
.4975
30°
.5000
.5025
.5050
.5075
.6100
.5125
31
.5150
.5175
.5200
.5225
.5250
.5275
32
.5299
.5324
.5348
.5373
.5398
.5422
33
.5446
.5471
.5495
.5519
.5544
.5568
34
.5592
.5616
.5640
.5664
.5688
.5712
35°
.5736
.5760
.5783
.5807
.5831
.5854
36
.5878
.5901
.5925
.5948
.5972
.5995
37
.6018
.6041
.6065
.6088
.6111
.6134
38
.6157
.6180
.6202
.6225
.6248
.6271
39
.6293
.6316
.6338
.6361
.6383
.6406
40°
.6428
.6450
.6472
.6494
.6517
.6539
41
.6561
.6583
.6604
.6626
.6648
.6670
42
.6691
.6713
.6734
.6756
.6777
,6799
43
.6820
.6841
.6862
.6884
.6905
.6926
44
.6947
.6967
.6988
.7009
.7030
.7050
45°
.7071
.7092
.7112
.7133
.7153
.7173
X
0'
10'
20'
30'
40'
50'
45°
.7071
.7092
.7112
.7133
.7153
.7173
46
.7193
.7214
.7234
.7254
.7274
.7294
47
.7314
.7333
.7353
.7373
.7392
.7412
48
.7431
.7451
.7470
.7490
.7509
.7528
49
.7547
.7566
.7585
.7604
.7623
.7642
50"
.7660
.7679
.7698
.7716
.7735
.7753
51
.7771
.7790
.7808
.7826
.7844
.7862
52
.7880
.7898
.7916
.7934
.7951
.7969
53
.7986
.8004
.8021
.8039
.8056
.8073
54
.8090
.8107
.8124
.8141
.8158
.8175
55°
.8192
.8208
.8225
.8241
.8258
.8274
56
.8290
.8307
.8323
.8339
.8355
.8371
57
.8387
.8403
.8418
.8434
.8450
.8465
58
.8480
.8496
.8511
.8526
.8542
.8557
59
.8572
.8587
.8G01
.8616
.8631
.8646
60°
.8660
.8675
.8689
.8704
.8718
.8732
61
.8746
.8760
.8774
.8788
.8802
.8816
62
.8829
.8843
.8857
.8870
.8884
.8897
63
.8910
.8923
.8936
.8949
.8962
.8975
64
.8988
.9001
.9013
.9026
.9038
.9051
65°
.9063
.9075
.9088
.9100
.9112
.9124
66
.9135
.9147
.9159
.9171
.9182
.9194
67
.9205
.9216
.9228
.9239
.9250
.9261
68
.9272
.9283
.9293
.9304
.9315
.9325
69
.9336
.9346
.9356
.9367
.9377
.9387
70°
.9397
.9407
.9417
.9426
.9436
.9446
71
.9455
.9465
.9474
.9483
.9492
.9502
72
.9511
.9520
.9528
.9537
.9546
.9555
73
.9563
.9572
.9580
.9588
.9596
.9605
74
.9613
.9621
.9628
.9636
.9644
.9652
75°
.9659
.9667
.9674
.9681
.9689
.9696
76
.9703
.9710
.9717
.9724
.9730
.9737
77
.9744
.9750
.9757
.9763
.9769
.9775
78
.9781
.9787
.9793
.9799
.9805
.9811
79
.9816
.9822
.9827
.9833
.9838
.9843
80°
.9848
.9853
.9858
.9863
.9868
.9872
81
.9877
.9881
.9886
.9890
.9894
.9899
82
.9903
.9907
.9911
.9914
.9918
.9922
83
.9925
.9929
.9932
.9936
.9939
.9942
84
.9945
.9948
.9951
.9954
.9957
.9959
85°
.9962
.9964
.9967
.9969
.9971
.9974
86
.9976
.9978
.9980
.9981
.9983
.9985
87
.9986
.9988
.9989
.9990
.9992
.9993
88
.9994
.9995
.9996
.9997
.9997
.9998
89
.9998
.9999
.9999
1.0000
1.0000
1.0000
90°
1.0000
206
0'
1
2
3
4
6
7
8
9
10 =
11
12
13
14
15 =
16
17
18
19
20°
21
22
23
24
25°
26
27
28
29
30°
31
32
33
34
35°
36
37
38
39
40°
41
42
43
44
0'
10'
20'
30'
1.0000 1.0000 1.0000 1.0000
.9998 .9998 .9997 .9997
.9994 .9993 .9992 .9990
.9986 .9985 .9983 .9981
.9976 .9974 .9971 .9969
.9962
.9945
.9926
.9903
.9877
.9848
.9816
.9781
.9744
.9703
.9659
.9613
.9563
.9611
.9455
.9397
.9336
.9272
.9205
.9135
.9063
.8988
.8910
.8829
.8746
.8660
.8572
.8480
.8387
.8290
.8192
.8090
.7986
.7880
.7771
.7660
.7547
.7431
.7314
.7193
.9959
.9942
.9922
.9899
.9872
.9843
.9811
.9775
.9737
.9696
.9652
.9605
.9555
.9502
.9446
.9387
.9325
.9261
.9194
.9124
.9051
.8975
.8897
.8816
.8732
.8646
.8557
.8465
.8371
.8274
.8175
.8073
.7969
.7862
.7753
.7642
.7528
.7412
.7294
.7173
.9957
.9939
.9918
.9894
.9868
.9838
.9805
.9769
.9730
.9689
.9644
.9596
.9546
.9492
.9436
.9377
.9315
.9250
.9182
.9112
.9038
.8962
.8884
.8802
.8718
.8631
.8542
.8450
.8355
.8258
.8158
.8056
.7951
.7844
.7735
.7623
.7509
.7392
.7274
.7153
.9954
.9936
.9914
.9890
.9863
.9833
.9799
.9763
.9724
.9681
.9630
.9588
.9537
.9483
.9426
.9367
.9304
.9239
.9171
.9100
.9026
.8949
.8870
.8788
.8704
.8616
.8526
.8434
.8339
.8241
.8141
.8039
.7934
.7826
.7716
.7604
.7490
.7373
.7254
.7133
40'
.9999
.9996
.9989
.9980
.9967
.9951
.9932
.9911
.9886
.9858
.9827
.9793
.9757
.9717
.9674
.9628
.9580
.9528
.9474
.9417
.9356
.9293
.9228
.9159
.9088
.9013
.8936
.8857
.8774
.8689
.8601
.8511
.8418
.8323
.8225
.8124
.8021
.7916
.7808
.7698
.7585
.7470
.7353
.7234
.7112
50'
.9999
.9995
.9988
.9978
.9964
.9948
.9929
.9907
.9881
.9853
.9822
.9787
.9750
.9710
.9667
.9621
.9572
.9520
.9465
.9407
.9346
.9283
.9216
.9147
.9075
.9001
.8923
.8843
.8760
.8675
.8587
.8496
.8403
.8307
.8208
.8107
.8004
.7898
.7790
.7679
.7566
.7451
.7333
.7214
.7092
45^
.7071 .7050 .7030 .7009 .6988 .6967
45"
46
47
48
49
50°
51
52
53
54
55°
56
57
58
59
60°
61
62
63
64
75'
76
77
78
79
80-^
81
82
83
84
85°
86
87
88
89
90'
0'
.7071
.6947
.6820
.6691
.6561
.6428
.6293
.6157
.6018
.5878
.5736
.5592
.5446
.5299
.5150
.5000
.4848
.4695
.4540
.4384
10'
.7050
.6926
.6799
.6670
.6539
.6406
.6271
.6134
.5995
.5854
.5712
.5568
.5422
.5275
.5125
.4975
.4823
.4669
.4514
.4358
20'
.7030
.6905
.6777
.6648
.6517
.6383
.6248
.6111
.5972
.5831
.5688
.5544
.5398
.5250
.5100
.4950
.4797
.4643
.4488
.4331
30'
.7009
.6884
.6756
.6626
.6494
.6361
.6225
.6088
.5948
.5807
.5664
.5519
.5373
.5225
.5075
.4924
.4772
.4617
.4462
.4305
40'
.6988
.6862
.6734
.6604
.6472
.6338
.6202
.6065
.5925
.5783
.5640
.5495
.5348
.5200
.5050
.4899
.4746
.4592
.4436
.4279
.2588
.2419
.2250
.2079
.1908
.1736
.1564
.1392
.1219
.1045
.0872
.0698
.0523
.0349
.0175
.2560
.2391
.2221
.2051
.1880
.1708
.1536
.1363
.1190
.1016
.0843
.0669
.0494
.0320
.0145
.2532
.2363
.2193
.2022
.1851
.1679
.1507
.1334
.1161
.0987
.0814
.0640
.0465
.0291
.0116
.2504
.2334
.2164
.1994
.1822
.1650
.1478
.1305
.1132
.0958
.0785
.0610
.0436
.0262
.0087
.2476
.2306
.2136
.1965
.1794
.1622
.1449
.1276
.1103
.0929
.0756
.0581
.0407
.0233
.0058
50'
.6967
.6841
.6713
.6583
.6450
.6316
.6180
.6041
.5901
.5760
.5616
.5471
.5324
.5175
.5025
.4874
.4720
.4566
.4410
.4253
65°
.4226
.4200
.4173
.4147
.4120
.4094
66
.4067
.4041
.4014
.3987
.3961
.3934
67
.3907
.3881
.3854
.3827
.3800
.3773
68
.3746
.3719
.3692
.3665
.3638
.3611
69
.3584
.3557
.3529
.3502
.3475
.3448
70°
.3420
.3393
.3365
.3338
.3311
.3283
71
.3256
.3228
.3201
.3173
.3145
.3118
72
.3090
.3062
.3035
.3007
.2979
.2952
73
.2924
.2896
.2868
.2840
.2812
.2784
74
.2756
.2728
.2700
.2672
.2644
.2616
.2447
.2278
.2108
.1937
.1765
.1593
.1421
.1248
.1074
.0901
.0727
.0552
.0378
.0204
.0029
.0000
207
0°
1
2
3
4
5°
6
7
8
9
10°
11
12
13
14
15°
16
17
18
19
20°
21
22
23
24
25°
26
27
28
29
30°
31
32
33
34
35°
36
37
38
39
40°
41
42
43
44
45'
0'
.0000
.0175
.0349
.0524
.0699
.0875
.1051
.1228
.1405
.1584
.1763
.1944
.2126
.2309
.2493
.2679
.2867
.3057
.3249
.3443
.3640
.3839
.4040
.4245
.4452
.4663
.4877
.5095
.5317
.5543
.5774
.6009
.6249
.6494
.6745
.7002
.7265
.7536
.7813
.8098
.8391
.8693
.9004
.9325
.9657
10'
.0029
.0204
.0378
.0553
.0729
.0904
.1080
.1257
.1435
.1614
.1793
.1974
.2156
.2339
.2524
.2711
.2899
.3089
.3281
.3476
.3673
.3872
.4074
.4279
.4487
.4699
.4913
.5132
.5354
.5581
.5812
.6048
.6289
.6536
.6787
.7046
.7310
.7581
.7860
.8146
.8441
.8744
.9057
.9380
.9713
20'
.0058
.0233
.0407
.0582
.0758
.0934
.1110
.1287
.1465
.1644
.1823
.2004
.2186
.2370
.2555
.2742
.2931
.3121
.3314
.3508
.3706
.3906
.4108
.4314
.4522
.4734
.4950
.5169
.5392
.5619
.5851
.6088
.6330
.6577
.6830
.7089
.7355
.7627
.7907
.8195
.8491
.8796
.9110
.9435
.9770
30'
.0087
.0262
.0437
.0612
.0787
.0963
.1139
.1317
.1495
.1673
.1853
.2035
.2217
.2401
.2586
.2773
.2962
.3153
.3346
.3541
.3739
.3939
.4142
.4348
.4557
.4770
.4986
.5206
.5430
.5668
.5890
.6128
.6371
.6619
.6873
.7133
.7400
.7673
.7954
.8243
.8541
.8847
.9163
.9490
.9827
40'
.0116
.0291
.0466
.0641
.0816
.0992
.1169
.1346
.1524
.1703
.1883
.2065
.2247
.2432
.2617
.2805
.2994
.3185
.3378
.3574
.3772
.3973
.4176
.4383
.4592
.4806
.5022
.5243
.5467
.5696
.5930
.6168
.6412
.6661
.6916
.7177
.7445
.7720
.8002
.8292
.8591
.8899
.9217
.9545
.9884
50'
.0145
.0320
.0495
.0670
.0846
.1022
.1198
.1376
.1554
.1733
.1914
.2095
.2278
.2462
.2648
.2836
.3026
.3217
.3411
.3607
.3805
.4006
.4210
.4417
.4628
.4841
.5059
.5280
.5505
.5735
.5969
.6208
.6453
.6703
.6959
.7221
.7490
.7766
.8050
.8342
.8642
.8952
.9271
.9601
.9942
1.0000 1.0058 1.0117 1.0176 1.0235 1.0295
45°
46
47
48
49
50°
51
52
53
54
55°
56
57
58
59
60°
61
62
63
64
65°
66
67
68
69
70°
71
72
73
74
75°
76
77
78
79
80°
81
82
83
84
85°
86
87
88
89
90°
0'
l.OOOO
1.0355
1.0724
1.1106
1.1504
1.1918
1.2349
1.2799
1.3270
1.3764
1.4281
1.4826
1.5399
1.6003
1.6643
1.7321
1.8040
1.8807
1.9626
2.0503
2.1445
2.2460
2.3559
2.4751
2.6051
2.7475
2.9042
3.0777
3.2709
3.4874
3.7321
4.0108
4.3315
4.7046
5.1446
5.6713
6.3138
7.1154
8.1443
9.5144
10'
1.0058
1.0416
1.0786
1.1171
1.1571
1.1988
1.2423
1.2876
1.3351
1.3848
1.4370
1.4919
1.5497
1.6107
1.6753
1.7437
1.8165
1.8940
1.9768
2.0655
2.1609
2.2637
2.3750
2.4960
2.6279
2.7725
2.9319
3.1084
3.3052
3.5261
3.7760
4.0611
4.3897
4.7729
5.2257
5.7694
6.4348
7.2687
8.3450
9.7882
11.430 11.826
14.301 14.924
19.081 20.206
28.636 31.242
57.290 68.750
20'
1.0117
1.0477
1.0850
1.1237
1.1640
1.2059
1.2497
1.2954
1.3432
1.3934
1.4460
1.5013
1.5597
1.6212
1.6864
1.7556
1.8291
1.9074
1.9912
2.0809
2.1775
2.2817
2.3945
2.5172
2.6511
2.7980
2.9600
3.1397
3.3402
3.5656
3.8208
4.1126
4.4494
4.8430
5.3093
5.8708
6.5606
7.4287
8.5555
10.078
12.251
15.605
21.470
34.368
85.940
30'
1.0176
1.0538
1.0913
1.1303
1.1708
1.2131
1.2572
1.3032
1.3514
1.4019
1.4550
1.5108
1.5697
1.6319
1.6977
1.7675
1.8418
1.9210
2.0057
2.0965
2.1943
2.2998
2.4142
2.5386
2.6746
2.8239
2.9887
3.1716
3.3759
3.6059
3.8667
4.1653
4.5107
4.9152
5.3955
5.9758
6.6912
7.5958
8.7769
10.385
12.706
16.350
22.904
38.188
114.59
40'
1.0235
1.0599
1.0977
1.1369
1.1778
1.2203
1.2647
1.3111
1.3597
1.4106
1.4641
1.5204
1.5798
1.6426
1.7090
1.7796
1.8546
1.9347
2.0204
2.1123
2.2113
2.3183
2.4342
2.5605
2.6985
2.8502
3.0178
3.2041
3.4124
3.6470
3.9136
4.2193
4.5736
4.9894
5.4845
6.0844
6.8269
7.7704
9.0098
10.712
50'
1.0295
1.0661
1.1041
1.1436
1.1847
1.2276
1.2723
1.3190
1.3680
1.4193
1.4733
1.5301
1.5900
1.6534
1.7205
1.7917
1.8676
1.9486
2.0353
2.1283
2.2286
2.3369
2.4545
2.5826
2.7228
2.8770
3.0475
3.2371
3.4495
3.6891
3.9617
4.2747
4.6382
5.0658
5.5764
6.1970
6.9682
7.9530
9.2553
11.059
13.197 13.727
17.169 18.075
24.542 26.432
42.964 49.104
171.89 343.77
208
0^
1
2
3
4
5°
6
7
8
9
10'
11
12
13
14
15 =
16
17
18
19
20°
21
22
23
24
25"
26
27
28
29
30 '^
31
32
33
34
35°
36
37
38
39
40°
41
42
43
44
0'
57.290
28.636
19.081
14.301
10'
343.77
49.104
26.432
18.075
13.727
11.430 11.059
9.5144 9.2553
8.1443 7.9530
7.1154 6.9682
6.3138 6.1970
20'
171.89
42.964
24.542
17.169
13.197
10.712
9.0098
7.7704
6.8269
6.0844
30'
114.59
38.188
22.904
1G.350
12.706
10.385
8.7769
7.5958
6.6912
5.9758
40'
85.940
34.368
21.470
15.605
12.251
50'
68.750
31.242
20.206
14.924
11.826
10.078 9.7882
8.5555 8.3450
7.4287 7.2687
6.5606 6.4348
5.8708 5.7694
5.6713 5,5764 5.4845 5.3955 5.3093 5.2257
5.1446 5.0658 4.9894 4.9152 4.8430 4.7729
4.7046 4.6382 4.5736 4.5107 4.4494 4.3897
4.3315 4.2747 4.2193 4.1653 4.1126 4.0611
4.0108 3.9617 3:9136 3.8667 3.8208 3.7760
45°
3.7321
3.6891
3.4874
3.4495
3.2709
3.2371
3.0777
3.0475
2.9042
2.8770
2.7475
2.7228
2.6051
2.5826
2.4751
2.4545
2.3559
2.3369
2.2460
2.2286
2.1445
2.0503
1.9626
1.8807
1.8040
1.7321
1.6643
1.6003
1.5399
1.4826
1.4281
1.3764
1.3270
1.2799
1.2349
1.1918
1.1504
1.1106
1.0724
1.0355
2.1283
2.0353
1.9486
1.8676
1.7917
1.7205
1.6534
1.5900
1.5301
1.4733
1.4193
1.3680
1.3190
1.2723
1.2276
1.1847
1.1436
1.1041
1.0661
1.0295
3.6470
3.4124
3.2041
3.0178
2.8502
2.6985
2.5605
2.4342
3.3183
2.2113
2.1123
2.0204
1.9347
1.8546
1.7796
1.7090
1.6426
1.5798
1.5204
1.4641
3.6059
3.3759
3.171G
2.9887
2.8239
2.6746
2.5386
2.4142
2.2998
2.1943
2.0965
2.0057
1.9210
1.8418
1.7675
1.6977
1.6319
1.5697
1.5108
1.4550
3.5656
3.3402
3.1397
2.9600
2.7980
2.6511
2.5172
2.3945
2.2817
2.1775
2.0809
1.9912
1.9074
1.8291
1.7556
1.6864
1.6212
1.5597
1.5013
1.4460
3.5261
3.3052
3.1084
2.9319
2.7725
2.6279
2.4960
2.3750
2.2637
2.1609
2.0655
1.9768
1.8940
1.8165
1.7437
1.6753
1.6107
1.5497
1.4919
1.4370
1.4106 1.4019 1.3934 1.3848
1.3597 1.3514 1.3432 1.3351
1.3111 1.3032 1.2954 1.2876
1.2647 1.2572 1.2497 1.2423
1.2203 1.2131 1.2059 1.1988
1.1778
1.1369
1.0977
1.0599
1.0235
1.1708
1.1303
1.0913
1.0538
1.0176
1.1640
1.1237
1.0850
1.0477
1.0117
1.1571
1.1171
1.0786
1.0416
1.0058
1.0000 .9942 .9884 .9827 .9770 .9713
45'
46
47
48
49
50=
51
52
53
54
55 '^
56
57
58
59
60'
61
62
63
64
65 =
66
67
68
69
70°
71
72
73
74
75^
76
77
78
79
80°
81
82
83
84
85 =
86
87
88
89
0'
10'
20'
30'
40'
50'
1.000 .9942 .9884 .9827 .9770 .9713
.9657 .9601 .9545 .9490 .9435 .9380
.9325 .9271 .9217 .9163 .9110 .9057
.9004 .8952 .8899 .8847 .8796 .8744
.8693 .8642 .8591 .8541 .8491 .8441
.8391 .8342 .8292 .8243 .8195 .8146
.8098 .8050 .8002 .7954 .7907 .7860
.7813 .7766 .7720 .7673 .7627 .7581
.7536 .7490 .7445 .7400 .7355 .7310
.7265 .7221 .7177 .7133 .7089 .7046
.7002 .6959
.6745 .6703
.6494 .6453
.6249 .6208
.6009 .5969
.5774 .5735
.5543 .5505
.5317 .5280
.5095 .5059
.4877 .4841
.4663
.4452
.4245
.4040
.3839
.3640
.3443
.3249
.3057
.2867
.2679
.2493
.2309
.2126
.1944
.1763
.1584
.1405
.1228
.1051
.0875
.0699
.0524
.0349
.0175
.4628
.4417
.4210
.4006
.3805
.3607
.3411
.3217
.3026
.2836
.2648
.2462
.2278
.2095
.1914
.1733
.1554
.1376
.1198
.1022
.0846
.0670
.0495
.0320
.0145
.6916
.6661
.6412
.6168
.5930
.5696
.5467
.5243
.5022
.4806
.4592
.4383
.4176
.3973
.3772
.3574
.3378
.3185
.2994
.2805
.2617
.2432
.2247
.2065
.1883
.1703
.1524
.1346
.1169
.0992
.0816
.0641
.0466
.0291
.0116
.6873
.6619
.6371
.6128
.5890
.5658
.5430
.5206
.4986
.4770
.4557
.4348
.4142
.3939
.3739
.3541
.3346
.3153
.2962
.2773
.2586
.2401
.2217
.2035
.1853
.1673
.1495
.1317
.1139
.0963
.0787
.0612
.0437
.0262
.0087
.6830
.6577
.6330
.6088
.5851
.5619
.5392
.5169
.4950
.4734
.4522
.4314
.4108
.3906
.3706
.3508
.3314
.3121
.2931
.2742
.2555
.2370
.2186
.2004
.1823
.1644
.1465
.1287
.1110
.0934
.0758
.0582
.0407
.0233
.0058
.6787
.6536
.6289
.6048
.5812
.5581
.5354
.5132
.4913
,4699
.4487
.4279
.4074
.3872
.3673
.3476
.3281
.3089
.2899
.2711
.2524
.2339
.2156
.1974
.1793
.1614
.1435
.1257
.1080
.0904
.0729
.0553
.0378
.0204
.0029
90° .0000
209
X
0'
10'
20'
30'
40'
50'
0°
1.000
1.000
1.000
1.000
1.000
1.000
1
1.000
1.000
1.000
1.000
1.000
1.001
2
1.001
1.001
1.001
1.001
1.001
1.001
3
1.001
1.002
1.002
1.002
1.002
1.002
4
1.002
1.003
1.003
1.003
1.003
1.004
5°
1.004
1.004
1.004
1.005
1.005
1.005
6
1.006
1.006
1.006
1.006
1.007
1.007
7
1.008
1.008
1.008
1.009
1.009
1.009
8
1.010
1.010
1.011
1.011
1.012
1.012
9
1.012
1.013
1.013
1.014
1.014
1.015
10"
1.015
1.016
1.016
1.017
1.018
1.018
11
1.019
1.019
1.020
1.020
1.021
1.022
12
1.022
1.023
1.024
1.024
1.025
1.026
13
1.026
1.027
1.028
1.028
1.029
1.030
14
1.031
1.031
1.032
1.033
1.034
1.034
15°
1.035
1.036
1.037
1.038
1.039
1.039
16
1.040
1.041
1.042
1.043
1.044
1.045
17
1.046
1.047
1.048
1.048
1.049
1.050
18
1.051
1.052
1.053
1.054
1.056
1.057
19
1.058
1.059
1.060
1.061
1.062
1.063
20°
1.064
1.065
1.066
1.068
1.069
1.070
21
1.071
1.072
1.074
1.075
1.076
1.077
22
1.079
1.080
1.081
1.082
1.084
1.085
23
1.086
1.088
1.089
1.090
1.092
1.093
24
1.095
1.096
1.097
1.099
1.100
1.102
25°
1.103
1.105
1.106
1.108
1.109
1.111
26
1.113
1.114
1.116
1.117
1.119
1.121
27
1.122
1.124
1.126
1.127
1.129
1.131
28
1.133
1.134
1.136
1.138
1.140
1.142
29
1.143
1.145
1.147
1.149
1.151
1.153
30°
1.155
1.157
1.159
1.161
1.163
1.165
31
1.167
1.169
1.171
1.173
1.175
1.177
32
1.179
1.181
1.184
1.186
1.188
1.190
33
1.192
1.195
1.197
1.199
1.202
1.204
34
1.206
1.209
1.211
1.213
1.216
1.218
35°
1.221
1.223
1.226
1.228
1.231
1.233
36
1.236
1.239
1,241
1.244
1.247
1.249
37
1.252
1.255
1.258
1.260
1.263
1.266
38
1.269
1.272
1.275
1.278
1.281
1.284
39
1.287
1.290
1.293
1.296
1.299
1.302
40°
1.305
1.309
1.312
1.315
1.318
1.322
41
1.325
1.328
1.332
1.335
1.339
1.342
42
1.346
1.349
1.353
1.356
1.360
1.364
43
1.367
1.371
1.375
1.379
1.382
1.386
44
1.390
1.394
1.398
1.402
1.406
1.410
45°
1.414
1.418
1.423
1.427
1.431
1.435
X
0'
10'
20'
30'
40'
50'
45°
1.414
1.418
1.423
1.427
1.431
1.435
46
1.440
1.444
1.448
1.453
1.457
1.462
47
1.466
1.471
1.476
1.480
1.485
1.490
48
1.494
1.499
1.504
1.509
1.514
1.519
49
1.524
1.529
1.535
1.510
1.545
1.550
50°
1.556
1.561
1.567
1.572
1.578
1.583
51
1.589
1.595
1.601
1.606
1.612
1.618
52
1.624
1.630
1.636
1.643
1.649
1.655
53
1.662
1.668
1.675
1.681
1.688
1.695
54
1.701
1.708
1.715
1.722
1.729
1.736
55°
1.743
1.751
1.758
1.766
1.773
1.781
56
1.788
1.796
1.804
1.812
1.820
1.828
57
1.836
1.844
1.853
1.861
1.870
1.878
58
1.887
1.896
1.905
1.914
1.923
1.932
59
1.942
1.951
1.961
1.970
1.980
1.990
60°
2.000
2.010
2.020
2.031
2.041
2.052
61
2.063
2.074
2.085
2.096
2.107
2.118
62
2.130
2.142
2.154
2.166
2.178
2.190
63
2.203
2.215
2.228
2.241
2.254
2.268
64
2.281
2.295
2.309
2.323
2.337
2.352
65°
2.366
2.381
2.396
2.411
2.427
2.443
66
2.459
2.475
2.491
2.508
2.525
2.542
67
2.559
2.577
2.595
2.613
2.632
2.650
68
2.669
2.689
2.709
2.729
2.749
2.769
69
2.790
2.812
2.833
2.855
2.878
2.901
70°
2.924
2.947
2.971
2.996
3.021
3.046
71
3.072
3.098
3.124
3.152
3.179
3.207
72
3.236
3.265
3.295
3.326
3.357
3.388
73
3.420
3.453
3.487
3.521
3.556
3.592
74
3.628
3.665
3.703
3.742
3.782
3.822
75°
3.864
3.906
3.950
3.994
4.039
4.086
76
4.134
4.182
4.232
4.284
4.336
4.390
77
4.445
4.502
4.560
4.620
4.682
4.745
78
4.810
4.876
4.945
5.016
5.089
5.164
79
5.241
5.320
5.403
5.487
5.575
5.665
80°
5.759
5.855
5.955
6.059
6.166
6.277
81
6.392
6.512
6.636
6.765
6.900
7.040
82
7.185
7.337
7.496
7.661
7.834
8.016
83
8.206
8.405
8.614
8.834
9.065
9.309
84
9.567
9.839
10.13
10.43
10.76
11.10
85°
11.47
11.87
12.29
12.75
13.23
13.76
86
14.34
14.96
15.64
16.38
17.20
18.10
87
19.11
20.23
21.49
22.93
24.56
26.45
88
28.65
31.26
34.38
38.20
42.98
49.11
89
57.30
68.76
85.95
114.6
171.9
343.8
90°
CO
210
0°
1
2
3
4
5°
6
7
8
9
10°
11
12
13
14
16
17
18
19
20°
21
22
23
24
0'
57.30
28.65
19.11
14.34
11.47
9.567
8.206
7.185
6.392
5.759
5.241
4.810
4.445
4.134
3.864
3.628
3.420
3.236
3.072
2.924
2.790
2.669
2.559
2.459
10'
343.8
49.11
26.45
18.10
13.76
11.10
9.309
8.016
7.040
6.277
5.665
5.164
4.745
4.390
4.086
3.822
3.592
3.388
3.207
3.046
2.901
2.769
2.650
2.542
2.443
20'
171.9
42.98
24.56
17.20
13.23
10.76
9.065
7.834
6.900
6.166
5.575
5.089
4.682
4.336
4.039
3.782
3.556
3.357
3.179
3.021
2.878
2.749
2.632
2.525
2.427
30'
114,6
38.20
22.93
16.38
12.75
10.43
8.834
7.661
6.765
6.059
5.487
5.016
4.620
4.284
3.994
3.742
3.521
3.326
3.152
2.996
2.855
2.729
2.613
2.508
2.411
40'
85.95
34.38
21.49
15.64
12.29
10.13
8.614
7.496
6.636
5.955
5.403
4.945
4.560
4.232
3.950
3.703
3.487
3.295
3.124
2.971
2.833
2.709
2.595
2.491
2.396
50'
68.76
31.26
20.23
14.96
11.87
9.839
8.405
7.337
6.512
5.855
5.320
4.876
4.502
4.182
3.906
3.665
3.453
3.265
3.098
2.947
2.812
2.689
2.577
2.475
2.381
25°
2.366
2.352
2.337
2.323
2.309
2.295
26
2.281
2.268
2.254
2.241
2.228
2.215
27
2.203
2.190
2.178
2.166
2.154
2.142
28
2.130
2.118
2.107
2.096
2.085
2.074
29
2.063
2.052
2.041
2.031
2.020
2.010
30°
2.000
1.990
1.980
1.970
1.961
1.951
31
1.942
1.932
1.923
1.914
1.905
1.896
32
1.887
1.878
1.870
1.861
1.853
1.844
33
1.836
1.828
1.820
1.812
1.804
1.796
34
1.788
1.781
1.773
1.766
1.758
1.751
35°
1.743
1.736
1.729
1.722
1.715
1.708
36
1.701
1.695
1.688
1.681
1.675
1.668
37
1.662
1.655
1.649
1.643
1.636
1.630
38
1.624
1.618
1.612
1.606
1.601
1.595
39
1.589
1.583
1.578
1.572
1.567
1.561
40°
1.556
1.550
1.545
1.540
1.535
1.529
41
1.524
1.519
1.514
1.509
1.504
1.499
42
1.494
1.490
1.485
1.480
1.476
1.471
43
1.466
1.462
1.457
1.453
1.448
1.444
44
1.440
1.435
1.431
1.427
1.423
1.418
45°
1.414
1.410
1.406
1.402
1.398
1.394
45°
46
47
48
49
50°
51
52
53
54
55°
56
57
58
59
60°
61
62
63
64
65°
66
67
68
69
70°
71
72
73
74
75°
76
77
78
79
80°
81
82
83
84
85°
86
87
88
89
90°
0'
1.414
1.390
1.367
1.346
1.325
1.305
1.287
1.269
1.252
1.236
1.221
1.206
1.192
1.179
1.167
1.103
1.095
1.086
1.079
1.071
1.064
1.058
1.051
1.046
1.040
1.035
1.031
1.026
1.022
1.019
1.015
1.012
1.010
1.008
1.006
1.004
1.002
1.001
1.001
1.000
1.000
10'
1.410
1.386
1.364
1.342
1.322
1.302
1.284
1.266
1.249
1.233
1.218
1.204
1.190
1.177
1.165
1.155 1.153
1.143 1.142
1.133 1.131
1.122 1.121
1.113 1.111
1.102
1.093
1.085
1.077
1.070
1.063
1.057
1.050
1.045
1.039
1.034
1.030
1.026
1.022
1.018
1.015
1.012
1.009
1.007
1.005
1.004
1.002
1.001
1.001
1.000
20'
1.406
1.382
1.360
1.339
1.318
1.299
1.281
1.263
1.247
1.231
1.216
1.202
1.188
1.175
1.163
1.151
1.140
1.129
1.119
1.109
1.100
1.092
1.084
1.076
1.069
1.062
1.056
1.049
1.044
1.039
1.034
1.029
1.025
1.021
1.018
1.014
1.012
1.009
1.007
1.005
1.003
1.002
1.001
1.000
1.000
30'
1.402
1.379
1.356
1.335
1.315
1.296
1.278
1.260
1.244
1.228
1.213
1.199
1.186
1.173
1.161
1.149
1.138
1.127
1.117
1.108
1.099
1.090
1.082
1.075
1.068
1.061
1.054
1.048
1.043
1.038
1.033
1.028
1.024
1.020
1.017
1.014
1.011
1.009
1.006
1.005
1.003
1.002
1.001
1.000
1.000
40'
1.398
1.375
1.353
1.332
1.312
1.293
1.275
1.258
1.241
1.226
1.211
1.197
1.184
1.171
1.159
1.097
1.089
1.081
1.074
1.066
1.060
1.053
1.048
1.042
1.037
1.032
1.028
1.024
1.020
1.016
1.013
1.011
1.008
1.006
1.004
1.003
1.002
1.001
1.000
1.000
50'
1.394
1.371
1.349
1.328
1.309
1.290
1.272
1.255
1.239
1.223
1.209
1.195
1.181
1.169
1.157
1.147 1.145
1.136 1.134
1.126 1.124
1.116 1.114
1.106 1.105
1.096
1.088
1.080
1.072
1.065
1.059
1.052
1.047
1.041
1.036
1.031
1.027
1.023
1.019
1.016
1.013
1.010
1.008
1.006
1.004
1.003
1.002
1.001
1.000
1.000
211
TABLE
9
NATURAL TRIGONOMETRIC FUNCTIONS (in radians)
X
Sin a;
Cosx
Tana:
Cot a;
Sec a;
Cscx
.00
.00000
1.00000
.00000
CO
1.00000
00
.01
.01000
.99995
.01000
99.9967
1.00005
100.0017
.02
.02000
.99980
.02000
49.9933
1.00020
50.0033
.03
.03000
.99955
.03001
33.3233
1.00045
33.3383
.04
,03999
.99920
.04002
24.9867
1.00080
25.0067
.05
.04998
.99875
.05004
19.9833
1.00125
20.0083
.06
.05996
.99820
.06007
16.6467
1.00180
16.6767
.07
.06994
.99755
.07011
14.2624
1.00246
14.2974
.08
.07991
.99680
.08017
12.4733
1.00321
12.5133
.09
.08988
.99595
.09024
11.0811
1.00406
11.1261
.10
.09983
.99500
.10033
9.9666
1.00502
10.0167
.11
.10978
.99396
.11045
9.0542
1.00608
9.1093
.12
.11971
.99281
.12058
8.2933
1.00724
8.3534
.13
.12963
.99156
.13074
7.6489
1.00851
7.7140
.14
.13954
.99022
.14092
7.0961
1.00988
7.1662
.15
.14944
.98877
.15114
6.6166
1.01136
6.6917
.16
.16932
.98723
.16188
6.1966
1.01294
6.2767
.17
.16918
.98558
.17166
5.8256
1.01463
5.9108
.18
.17903
.98384
.18197
6.4954
1.01642
5.5857
.19
.18886
.98200
.19232
6.1997
1.01833
5.2950
.20
.19867
.98007
.20271
4.9332
1.02034
5.0335
.21
.20846
.97803
.21314
4.6917
1.02246
4.7971
.22
.21823
.97590
.22362
4.4719
1.02470
4.5823
.23
.22798
.97367
.23414
4.2709
1.02705
4.3864
.24
.23770
.97134
.24472
4.0864
1.02951
4.2069
.25
.24740
.96891
.25534
3.9163
1.03209
4.0420
.26
.25708
.96639
.26602
3.7591
1.03478
3.8898
.27
.26673
.96377
.27676
3.6133
1.03759
3.7491
.28
.27636
.96106
.28755
3.4776
1.04052
3.6185
.29
.28595
.95824
.29841
3.3511
1.04358
3.4971
.30
.29552
.95534
.30934
3.2327
1.04675
3.3839
.31
.30506
.95233
.32033
3.1218
1.05005
3.2781
.32
.31457
.94924
.33139
3.0176
1.05348
3.1790
.33
.32404
.94604
.34252
2.9195
1.05704
3.0860
.34
.33349
.94275
.35374
2.8270
1.06072
2.9986
.35
.34290
.93937
.36503
2.7395
1.06454
2.9163
.36
.35227
.93590
.37640
2.6567
1.06849
2.8387
.37
.36162
.93233
.38786
2.5782
1.07258
2.7654
.38
.37092
.92866
.39941
2.5037
1.07682
2.6960
.39
.38019
.92491
.41105
2.4328
1.08119
2.6303
.40
.38942
.92106
.42279
2.3652
1.08570
2.5679
212
Toble 9
(continued)
NATURAL TRIGONOMETRIC FUNCTIONS (in radians)
X
Sin a;
Cos a;
Tan re
Cot a;
Sec a;
Csca;
-
.40
.38942
.92106
.42279
2.3652
1.0857
2.5679
.41
.39861
.91712
.43463
2.3008
1.0904
2.5087
.42
.40776
.91309
.44657
2.2393
1.0952
2.4524
.43
.4X687
.90897
.45862
2.1804
1.1002
2.3988
.44
.42594
.90475
.47078
2.1241
1.1053
2.3478
.45
.43497
.90045
.48306
2.0702
1.1106
2.2990
.46
.44395
.89605
.49545
2.0184
1.1160
2.2525
.47
.45289
.89157
.50797
1.9686
1.1216
2.2081
.48
.46178
.88699
.52061
1.9208
1.1274
2.1655
.49
.47063
.88233
.53339
1.8748
1.1334
2.1248
.50
.47943
.87758
.54630
1.8305
1.1395
2.0858
.51
.48818
.87274
.55936
1.7878
1.1458
2.0484
.52
.49688
.86782
.57256
1.7465
1.1523
2.0126
.53
.50553
.86281
.58592
1.7067
1.1590
1.9781
.54
.51414
.85771
.59943
1.6683
1.1659
1.9450
.55
.52269
.85252
.61311
1.6310
1.1730
1.9132
.56
.53119
.84726
.62695
1.5950
1.1803
1.8826
.57
.53963
.84190
.64097
1.5601
1.1878
1.8531
.58
.54802
.83646
.65517
1.5263
1.1955
1.8247
.59
.55636
.83094
.66956
1.4935
1.2035
1.7974
.60
.56464
.82534
.68414
1.4617
1.2116
1.7710
.61
.67287
.81965
.69892
1.4308
1.2200
1.7456
.62
.58104
.81388
.71391
1.4007
1.2287
1.7211
.63
.58914
.80803
.72911
1.3715
1.2376
1.6974
.64
.59720
.80210
.74454
1.3431
1.2467
1.6745
.65
.60519
.79608
.76020
1.3154
1.2561
1.6524
.66
.61312
.78999
.77610
1.2885
1.2658
1.6310
.67
.62099
.78382
.79225
1.2622
1.2758
1.6103
.68
.62879
.77757
.80866
1.2366
1.2861
1.5903
.69
.63654
.77125
.82534
1.2116
1.2966
1.5710
.70
.64422
.76484
.84229
1.1872
1.3075
1.5523
.71
.65183
.75836
.85953
1.1634
1.3186
1.5341
.72
.65938
.75181
.87707
1.1402
1.3301
1.5166
.73
.66687
.74517
.89492
1.1174
1.3420
1.4995
.74
.67429
.73847
.91309
1.0952
1.3542
1.4830
.75
.68164
.73169
.93160
1.0734
1.3667
1.4671
.76
.68892
.72484
.95045
1.0521
1.3796
1.4515
.77
.69614
.71791
.96967
1.0313
1.3929
1.4365
.78
.70328
.71091
.98926
1.0109
1.4066
1.4219
.79
.71035
.70385
1.0092
.99084
1.4208
1.4078
-
.80
.71736
.69671
1.0296
.97121
1.4353
1.3940
213
Table?
(continued)
NATURAL TRIGONOMETRIC FUNCTIONS (in radians)
X
Sinx
Cos a;
Tana:
CotK
Sec X
Csc X
.80
.71736
.69671
1.0296
.97121
1.4353
1.3940
.81
.72429
.68950
1.0505
.95197
1.4503
1.3807
.82
.73115
.68222
1.0717
.93309
1.4658
1.3677
.83
.73793
.67488
1.0934
.91455
1.4818
1.3551
.84
.74464
.66746
1.1156
.89635
1.4982
1.3429
.85
.75128
.65998
1.1383
.87848
1.5152
1.3311
.86
.75784
.65244
1.1616
.86091
1.5327
1.3195
.87
.76433
.64483
1.1853
.84365
1.5508
1.3083
.88
.77074
.63715
1.2097
.82668
1.5695
1.2975
.89
.77707
.62941
1.2346
.80998
1.5888
1.2869
.90
.78333
.62161
1.2602
.79355
1.6087
1.2766
.91
.78950
.61375
1.2864
.77738
1.6293
1.2666
.92
.79560
.60582
1.3133
.76146
1.6507
1.2569
.93
.80162
.59783
1.3409
.74578
1.6727
1.2475
.94
.80756
.58979
1.3692
.73034
1.6955
1.2383
.95
.81342
.68168
1.3984
.71511
1.7191
1.2294
.96
.81919
.57352
1.4284
.70010
1.7436
1.2207
.97
.82489
.56530
1.4592
.68531
1.7690
1.2123
.98
.83050
.55702
1.4910
.67071
1.7953
1.2041
.99
.83603
.54869
1.5237
.65631
1.8225
1.1961
1.00
.84147
.54030
1.5574
.64209
1.8508
1.1884
1.01
.84683
.53186
1.5922
.62806
1.8802
1.1809
1.02
.85211
.52337
1.6281
.61420
1.9107
1.1736
1.03
.85730
.51482
1.6652
.60051
1.9424
1.1665
1.04
.86240
.50622
1.7036
.58699
1.9754
1.1595
1.05
.86742
.49757
1.7433
.57362
2.0098
1.1528
1.06
.87236
.48887
1.7844
.56040
2.0455
1.1463
1.07
.87720
.48012
1.8270
.54734
2.0828
1.1400
1.08
.88196
.47133
1.8712
.53441
2.1217
1.1338
1,09
.88663
.46249
1.9171
.52162
2.1622
1.1279
1.10
.89121
.45360
1.9648
.50897
2.2046
1.1221
1.11
.89570
.44466
2.0143
.49644
2.2489
1.1164
1.12
.90010
.43568
2.066O
.48404
2.2952
1.1110
1.13
.90441
.42666
2.1198
.47175
2.3438
1.1057
1.14
.90863
.41759
2.1759
.45959
2.3947
1.1006
1.15
.91276
.40849
2.2346
.44753
2.4481
1.0956
1.16
.91680
.39934
2.2958
.43558
2.5041
1.0907
1.17
.92075
.39015
2.3600
.42373
2.5631
1.0861
1.18
.92461
.38092
2.4273
.41199
2.6252
1.0815
1.19
.92837
.37166
2.4979
.40034
2.6906
1.0772
1.20
.93204
.36236
2.5722
.38878
2.7597
1.0729
214
Table 9
(continued)
NATURAL TRIGONOMETRIC FUNCTIONS (in radians!
X
Sin X
Cos a;
Tana:
Cot a;
Sec a;
Csca;
1.20
.93204
.36236
2.5722
.38878
2.7597
1.07292
1.21
.93562
.35302
2.6503
.37731
2.8327
1.06881
1.22
.93910
.34365
2.7328
.36593
2.9100
1.06485
1.23
.94249
.33424
2.8198
.35463
2.9919
1.06102
1.24
.94578
.32480
2.9119
.34341
3.0789
1.05732
1.25
.94898
.31532
3.0096
.33227
3.1714
1.05376
1.26
.95209
.30582
3.1133
.32121
3.2699
1.05032
1.27
.95510
.29628
3.2236
.31021
3.3752
1.04701
1.28
.95802
.28672
3.3414
.29928
3.4878
1.04382
1.29
.96084
.27712
3.4672
.28842
3.6085
1.04076
1.30
.96356
.26750
3.6021
.27762
3.7383
1.03782
1.31
.96618
.25785
3.7471
.26687
3.8782
1.03500
1.32
.96872
.24818
3.9033
.25619
4.0294
1.03230
1.33
.97115
.23848
4.0723
.24556
4.1933
1.02971
1.34
.97348
.22875
4.2556
.23498
4.3715
1.02724
1.35
.97572
.21901
4.4552
.22446
4.5661
1.02488
1.36
.97786
.20924
4.6734
.21398
4.7792
1.02264
1.37
.97991
.19945
4,9131
.20354
5.0138
1.02050
1.38
.98185
.18964
5.1774
.19315
5.2731
1.01848
1.39
.98370
.17981
5.4707
.18279
5.5613
1.01657
1.40
.98545
.16997
5.7979
.17248
5.8835
1.01477
1.41
.98710
.16010
6.1654
.16220
6.2459
1.01307
1.42
.98865
.15023
6.5811
.15195
6.6567
1.01148
1.43
.99010
.14033
7.0555
.14173
7.1260
1.00999
1.44
.99146
.13042
7.6018
.13155
7.6673
1.00862
1.45
.99271
.12050
8.2381
.12139
8.2986
1.00734
1.46
.99387
.11057
8.9886
.11125
9.0441
1.00617
1.47
.99492
.10063
9.8874
.10114
9.9378
1.00510
1.48
.99588
.09067
10.9834
.09105
11.0288
1.00414
1.49
.99674
.08071
12.3499
.08097
12.3903
1.00327
1.50
.99749
.07074
14.1014
.07091
14.1368
1.00251
1.51
.99815
.06076
16.4281
.06087
16.4585
1.00185
1.52
.99871
.05077
19.6695
.05084
19.6949
1.00129
1.53
.99917
.04079
24.4984
.04082
24.5188
1.00083
1.54
.99953
.03079
32.4611
.03081
32.4765
1.00047
1.55
.99978
.02079
48.0785
.02080
48.0889
1.00022
1.56
.99994
.01080
92.6205
.01080
92.6259
1.00006
1.57
1.00000
.00080
1255.77
.00080
1255.77
1.00000
1.58
.99996
-.00920
-108.649
-.00920
-108.654
1.00004
1.59
.99982
-.01920
-52.0670
-.01921
-52.0766
1.00018
1.60
.99957
-.02920
-34.2325
-.02921
-34.2471
1.00043
215
TABLE
10
log sin X (a: in degrees and minutes)
[subtract 10 from each entry]
X
0'
10'
20'
30'
40'
50'
0°
_
7.4637
7.7648
7.9408
8.0658
8.1627
1
8.2419
8.3088
8.3668
8.4179
8.4637
8.5050
2
8.5428
8.5776
8.6097
8.6397
8.6677
8.6940
3
8.7188
8.7423
8.7645
8.7857
8.8059
8.8251
4
8.8436
8.8613
8.8783
8.8946
8.9104
8.925G
5-
8.9403
8.9545
8.9682
8.9816
8.9945
9.0070
6
9.0192
9.0311
9.0426
9.0539
9.0648
9.0755
7
9.0859
9.0961
9.1060
9.1157
9.1252
9.1345
8
9.1436
9.1525
9.1612
9.1697
9.1781
9.1863
9
9.1943
9.2022
9.2100
9.2176
9.2251
9.2324
10°
9.2397
9.2468
9.2538
9.2606
9.2674
9.2740
11
9.2806
9.2870
9.2934
9.2997
9.3058
9.3119
12
9.3179
9.3238
9.3296
9.3353
9.3410
9.3466
13
9.3521
9.3575
9.3629
9.3682
9.3734
9.3786
14
9.3837
9.3887
9.3937
9.3986
9.4035
9.4083
16°
9.4130
9.4177
9.4223
9.4269
9.4314
9.4359
16
9.4403
9.4447
9.4491
9.4533
9.4576
9.4618
17
9.4659
9.4700
9.4741
9.4781
9.4821
9.4861
18
9.4900
9.4939
9.4977
9.5015
9.5052
9.5090
19
9.5126
9.5163
9.5199
9.5235
9.5270
9.5306
20°
9.5341
9.5375
9.5409
9.5443
9.5477
9.5510
21
9.5543
9.5576
9.5609
9.5641
9.5673
9.5704
22
9.5736
9.5767
9.5798
9.5828
9.5859
9.5889
23
9.5919
9.5948
9.5978
9.6007
9.6036
9.6065
24
9.6093
9.6121
9.6149
9.6177
9.6205
9.6232
25°
9.6259
9.6286
9.6313
9.6340
9.6366
9.6392
26
9.6418
9.6444
9.6470
9.6495
9.6521
9.6546
27
9.6570
9.6595
9.6620
9.6644
9.6668
9.6692
28
9.6716
9.6740
9.6763
9.6787
9.6810
9.6833
29
9.6856
9.6878
9.6901
9.6923
9.6946
9.6968
30°
9.6990
9.7012
9.7033
9.7055
9.7076
9.7097
31
9.7118
9.7139
9.7160
9.7181
9.7201
9.7222
32
9.7242
9.7262
9.7282
9.7302
9.7322
9.7342
33
9.7361
9.7380
9.7400
9.7419
9.7438
9.7457
34
9.7476
9.7494
9.7513
9.7531
9.7550
9.7568
35°
9.7586
9.7604
9.7622
9.7640
9.7657
9.7675
36
9.7692
9.7710
9.7727
9.7744
9.7761
9.7778
37
9.7795
9.7811
9.7828
9.7844
9.7861
9.7877
38
9.7893
9.7910
9.7926
9.7941
9.7957
9.7973
39
9.7989
9.8004
9.8020
9.8035
9.8050
9.8066
40''
9.8081
9.8096
9.8111
9.8125
9.8140
9.8155
41
9.8169
9.8184
9.8198
9.8213
9.8227
9.8241
42
9.8255
9.8269
9.8283
9.8297
9.8311
9.8324
43
9.8338
9.8351
9.8365
9.8378
9.8391
9.8405
44
9.8418
9.8431
9.8444
9.8457
9.8469
9.8482
45'-'
9.8495
9.8507
9.8520
9.8532
9.8545
9.8557
216
Table 10
(continued)
lOQ sin X {x in degrees and minutes)
[subtract 10 from each entry]
X
0'
10'
20'
30'
40'
50'
45°
9.8495
9.8507
9.8520
9.8532
9.8545
9.8557
46
9.8569
9.8582
9.8594
9.8606
9.8618
9.8629
47
9.8641
9.8653
9.8665
9.8676
9.8688
9.8699
48
9.8711
9.8722
9.8733
9.8745
9.8756
9.8767
49
9.8778
9.8789
9.8800
9.8810
9.8821
9.8832
50°
9.8843
9.8853
9.8864
9.8874
9.8884
9.8895
51
9.8905
9.8915
9.8925
9.8935
9.8945
3.8955
52
9.8965
9.8975
9.8985
9.8995
9.9004
9.9014
53
9.9023
9.9033
9.9042
9.9052
9.9061
9.9070
54
9.9080
9.9089
9.9098
9.9107
9.9116
9.9125
55°
9.9134
9.9142
9.9151
9.9160
9.9169
9.9177
56
9.9186
9.9194
9.9203
9.9211
9.9219
9.9228
57
9.9236
9.9244
9.9252
9.9260
9.9268
9.9276
58
9.9284
9.9292
9.9300
9.9308
9.9315
9.9323
59
9.9331
9.9338
9.9346
9.9353
9.9361
9.9368
60°
9.9375
9.9383
9.9390
9.9397
9.9404
9.9411
61
9.9418
9.9425
9.9432
9.9439
9.9446
9.9453
62
9.9459
9.9466
9.9473
9.9479
9.9486
9.9492
63
9.9499
9.9505
9.9512
9.9518
9.9524
9.9530
64
9.9537
9.9543
9.9549
9.9555
9.9561
9.9567
65°
9.9573
9.9579
9.9584
9.9590
9.9596
9.9602
66
9.9607
9.9613
9.9618
9.9624
9.9629
9.9635
67
9.9640
9.9646
9.9651
9.9656
9.9661
9.9667
68
9.9672
9.9677
9.9682
9.9687
9.9692
9.9697
69
9.9702
9.9706
9.9711
9.9716
9.9721
9.9725
70°
9.9730
9.9734
9.9739
9.9743
9.9748
9.9752
71
9.9757
9.9761
9.9765
9.9770
9.9774
9,9778
72
9.9782
9.9786
9.9790
9.9794
9.9798
9.9802
73
9.9806
9.9810
9.9814
9.9817
9.9821
9.9825
74
9.9828
9.9832
9.9836
9.9839
9.9843
9.9846
75°
9.9849
9.9853
9.9856
9.9859
9.9863
9.9866
76
9.9869
9.9872
9.9875
9.9878
9.9881
9.9884
77
9.9887
9.9890
9.9893
9.9896
9.9899
9.9901
78
9.9904
9.9907
9.9909
9.9912
9.9914
9.9917
79
9.9919
9.9922
9.9924
9.9927
9.9929
9.9931
80°
9.9934
9.9936
9.9938
9.9940
9.9942
9.9944
81
9.9946
9.9948
9.9950
9.9952
9.9954
9.9956
82
9.9958
9.9959
9.9961
9.9963
9.9964
9.9966
83
9.9968
9.9969
9.9971
9.9972
9.9973
9.9975
84
9.9976
9.9977
9.9979
9.9980
9.9981
9.9982
85°
9.9983
9.9985
9.9986
9.9987
9.9988
9.9989
86
9.9989
9.9990
9.9991
9.9992
9.9993
9.9993
87
9.9994
9.9995
9.9995
9.9996
9.9996
9.9997
88
9.9997
9.9998
9.9998
9.9999
9.9999
9.9999
89
9.9999
10.0000
10.0000
10.0000
10.0000
10.0000
90°
10.0000
217
TABLE
n
log COS X {x in degrees and minutes)
[subtract 10 from each entry]
X
0'
10'
20'
30'
40'
50'
0^
10.0000
10.0000
10.0000
10.0000
10.0000
10.0000
1
9.9999
9.9999
9.9999
9.9999
9.9998
9.9998
2
9.9997
9.9997
9.9996
9.9996
9.9995
9.9995
3
9.9994
9.9993
9.9993
9.9992
9.9991
9.9990
4
9.9989
9.9989
9.9988
9.9987
9.9986
9.9985
5°
9.9983
9.9982
9.9981
9.9980
9.9979
9.9977
6
9.9976
9.9975
9.9973
9.9972
9.9971
9.9969
7
9.9968
9.9966
9.9964
9.9963
9.9961
9.9959
8
9.9958
9.9956
9.9954
9.9952
9.9950
9.9948
9
9.9946
9.9944
9.9942
9.9940
9.9938
9.9936
10°
9.9934
9.9931
9.9929
9.9927
9.9924
9.9922
11
9.9919
9.9917
9.9914
9.9912
9.9909
9.9907
12
9.9904
9.9901
9.9899
9.9896
9.9893
9.9890
13
9.9887
9.9884
9.9881
9.9878
9.9875
9.9872
14
9.9869
9.9866
9.9863
9.9859
9.9856
9.9853
15°
9.9849
9.9846
9.9843
9.9839
9.9836
9.9832
16
9.9828
9.9825
9.9821
9.9817
9.9814
9.9810
17
9.9806
9.9802
9.9798
9.9794
9.9790
9.9786
18
9.9782
9.9778
9.9774
9.9770
9.9765
9.9761
19
9.9757
9.9752
9.9748
9.9743
9.9739
9.9734
20°
9.9730
9.9725
9.9721
9.9716
9.9711
9.9706
21
9.9702
9.9697
9.9692
9.9687
9.9682
9.9677
22
9.9672
9.9667
9.9661
9.9656
9.9651
9.9646
23
9.9640
9.9635
9.9629
9.9624
9.9618
9.9613
24
9.9607
9.9602
9.9596
9.9590
9.9584
9.9579
25°
9.9573
9.9567
9.9561
9.9555
9.9549
9,9543
26
9.9537
9.9530
9.9524
9.9518
9.9512
9.9505
27
9.9499
9.9492
9.9486
9.9479
9.9473
9.9466
28
9.9459
9.9453
9.9446
9.9439
9.9432
9.9425
29
9.9418
9.9411
9.9404
9.9397
9.9390
9.9383
30°
9.9375
9.9368
9.9361
9.9353
9.9346
9.9338
31
9.9331
9.9323
9.9315
9.9308
9.9300
9.9292
32
9.9284
9.9276
9.9268
9.9260
9.9252
9.9244
33
9.9236
9.9228
9.9219
9.9211
9.9203
9.9194
34
9.9186
9.9177
9.9169
9.9160
9.9151
9.9142
35°
9.9134
9.9125
9.9116
9.9107
9.9098
9.9089
36
9.9080
9.9070
9.9061
9.9052
9.9042
9.9033
37
9.9023
9.9014
9.9004
9.8995
9.8985
9.8975
38
9.8965
9.8955
9.8945
9.8935
9.8925
9.8915
39
9.8905
9.8895
9.8884
9.8874
9.8864
9.8853
40°
9.8843
9.8832
9.8821
9.8810
9.8800
9.8789
41
9.8778
9.8767
9.8756
9.8745
9.8733
9.8722
42
9.8711
9.8699
9.8688
9.8676
9.8665
9.8653
43
9.8641
9.8629
9.8618
9.8606
9.8594
9.8582
44
9.8569
9.8557
9.8545
9.8532
9.8520
9.8507
45°
9.8495
9.8482
9.8469
9.8457
9.8444
9.8431
218
Table 1 1
(continued)
■Og cos X {x in degrees and minutes)
[subtract 10 from each entryl
X
0'
10'
20'
30'
40'
50'
45°
9.8495
9.8482
9.8469
9,8457
9.8444
9.8431
46
9.8418
9.8405
9.8391
9.8378
9.8365
9.8351
47
9.8338
9.8324
9.8311
9.8297
9.8283
9.8269
48
9.8255
9.8241
9.8227
9.8213
9.8198
9.8184
49
9.81G9
9.8155
9.8140
9.8125
9.8111
9.8096
50°
9.8081
9.8066
9.8050
9.8035
9.8020
9.8004
51
9.7989
9.7973
9.7957
9.7941
9.7926
9.7910
52
9.7893
9.7877
9.7861
9.7844
9.7828
9.7811
53
9.7795
9.7778
9.7761
9.7744
9.7727
9.7710
54
9.7692
9.7675
9.7657
9.7640
9.7622
9.7604
55=^
9.7586
9.7568
9.7550
9.7531
9.7513
9.7494
56
9.7476
9.7457
9.7438
9.7419
9.7400
9.7380
57
9.7361
9.7342
9.7322
9.7302
9.7282
9.7262
58
9.7242
9.7222
9.7201
9.7181
9.7160
9.7139
59
9.7118
9.7097
9.7076
9.7055
9.7033
9.7012
60°
9.6990
9.6968
9.6946
9.6923
9.6901
9.6878
61
9.6856
9.6833
9.6810
9.6787
9.6763
9.6740
62
9.6716
9.6692
9.6668
9.6644
9.6620
9.6595
63
9.6570
9.6546
9.6521
9.6495
9.6470
9.6444
64
9.6418
9.6392
9.6366
9.6340
9.6313
9.6286
65°
9.6259
9.6232
9.6205
9.6177
9.6149
9.6121
66
9.6093
9.6065
9.6036
9.6007
9.5978
9.5948
67
9.5919
9.5889
9.5859
9.5828
9.5798
9.5767
68
9.5736
9.5704
9.5673
9.5641
9.5609
9.5576
69
9.5543
9.5510
9.5477
9.5443
9.5409
9.5375
70°
9.5341
9.5306
9.5270
9.5235
9.5199
9.5163
71
9.5126
9.5090
9.5052
9.5015
9.4977
9.4939
72
9.4900
9.4861
9.4821
9.4781
9.4741
9.4700
73
9.4659
9.4618
9.4576
9.4533
9.4491
9.4447
74
9.4403
9.4359
9.4314
9.4269
9.4223
9.4177
75°
9.4130
9.4083
9.4035
9.3986
9.3937
9.3887
76
9.3837
9.3786
9.3734
9.3682
9.3629
9.3575
77
9.3521
9.3466
9.3410
9.3353
9.3296
9.3238
78
9.3179
9.3119
9.3058
9.2997
9.2934
9.2870
79
9.2806
9.2740
9.2674
9.2606
9.2538
9.2468
80°
9.2397
9.2324
9.2251
9.2176
9.2100
9.2022
81
9.1943
9.1863
9.1781
9.1697
9.1612
9.1525
82
9.1436
9.1345
9.1252
9.1157
9.1060
9.0961
83
9.0859
9.0756
9.0648
9.0539
9.0426
9.0311
84
9.0192
9.0070
8.9945
8.9816
8.9682
8.9545
85°
8.9403
8.9256
8.9104
8.8946
8.8783
8.8613
86
8.8436
8.8251
8.8059
8.7857
8.7645
8.7423
87
8.7188
8.6940
8.6677
8.6397
8.6097
8.5776
88
8.5428
8.5050
8.4637
8.4179
8.3668
8.3088
89
8.2419
8.1627
8.0658
7.9408
7.7648
7.4637
90°
-
219
TABLE
12
log ton X (a; in degrees and minutes)
[subtract 10 from each entry]
X
0'
10'
20'
30'
40'
50'
0°
—
7.4637
7.7648
7.9409
8.0658
8.1627
1
8.2419
8.3089
8.3669
8.4181
8.4638
8.5053
2
8.5431
8.5779
8.6101
8.6401
8.6682
8.6945
3
8.7194
8.7429
8.7652
8.7865
8.8067
8.8261
4
8.8446
8.8624
8.8795
8.8960
8.9118
8.9272
5°
8.9420
8.9563
8.9701
8.9836
8.9966
9.0093
6
9.0216
9.0336
9.0453
9.0567
9.0678
9.0786
7
9.0891
9.0995
9.1096
9.1194
9.1291
9.1385
8
9.1478
9.1569
9.1658
9.1745
9.1831
9.1915
9
9.1997
9.2078
9.2158
9.2236
9.2313
9.2389
10°
9.2463
9.2536
9.2609
9.2680
9.2750
9.2819
11
9.2887
9.2953
9.3020
9.3085
9.3149
9.3212
12
9.3275
9.3336
9.3397
9.3458
9.3517
9.3576
13
9.3634
9.3691
9.3748
9.3804
9.3859
9.3914
14
9.3968
9.4021
9.4074
9.4127
9.4178
9.4230
15°
9.4281
9.4331
9.4381
9.4430
9.4479
9.4527
16
9.4575
9.4622
9.4669
9.4716
9.4762
9.4808
17
9.4853
9.4898
9.4943
9.4987
9.5031
9.5075
18
9.5118
9.5161
9.5203
9.5245
9.52S7
9.5329
19
9.5370
9.5411
9.5451
9.5491
9.5531
9.5571
20°
9.5611
9.5650
9.5689
9.5727
9.5766
9.5804
21
9.5842
9.5879
9.5917
9.5954
9.5991
9.6028
22
9.6064
9.6100
9.6136
9.6172
9.6208
9.6243
23
9.6279
9.6314
9.6348
9.6383
9.6417
9.6452
24
9.6486
9.6520
9.6553
9.6587
9.6620
9.6654
25°
9.6687
9.6720
9.6752
9.6785
9.6817
9.6850
26
9.6882
9.6914
9.6946
9.6977
9.7009
9.7040
27
9.7072
9.7103
9.7134
9.7165
9.7196
9.7226
28
9.7257
9.7287
9.7317
9.7348
9.7378
9.7408
29
9.7438
9.7467
9.7497
9.7526
9.7556
9.7585
30"
9.7614
9.7644
9.7673
9.7701
9.7730
9.7759
31
9.7788
9.7816
9.7845
9.7873
9.7902
9.7930
32
9.7958
9.7986
9.8014
9.8042
9.8070
9.8097
33
9.8125
9.8153
9.8180
9.8208
9.8235
9.8263
34
9.8290
9.8317
9.8344
9.8371
9.8398
9.8425
35°
9.8452
9.8479
9.8506
9.8533
9.8559
9.8586
36
9.8613
9.8639
9.8666
9.8692
9.8718
9.8745
37
9.8771
9.8797
9.8824
9.8850
9.8876
9.8902
38
9.8928
9.8954
9.8980
9.9006
9.9032
9.9058
39
9.9084
9.9110
9.9135
9.9161
9.9187
9.9212
40°
9.9238
9.9264
9.9289
9.9315
9.9341
9.9366
41
9.9392
9.9417
9.9443
9.9468
9.9494
9.9619
42
9.9544
9.9570
9.9595
9.9621
9.9646
9.9671
43
9.9697
9.9722
9.9747
9.9772
9.9798
9.9823
44
9.9848
9.9874
9.9899
9.9924
9.9949
9.9975
45°
10.0000
10.0025
10.0051
10.0076
10.0101
10.0126
220
Table 12
(continued)
log ton X (a; in degrees and minutes)
[subtract 10 from each entry]
X
0'
10'
20'
30'
40'
50'
45°
10.0000
10.0025
10.0051
10.0076
10.0101
10.0126
46
10.0152
10.0177
10.0202
10.0228
10.0253
10.0278
47
10.0303
10.0329
10.0354
10.0379
10.0405
10.0430
48
10.0456
10.0481
10.0506
10.0532
10.0557
10.0583
49
10.0608
10.0634
10.0659
10.0685
10.0711
10.0736
50°
10.0762
10.0788
10.0813
10.0839
10.0865
10.0890
51
10.0916
10.0942
10.0968
10.0994
10.1020
10.1046
52
10.1072
10.1098
10.1124
10.1150
10.1176
10.1203
53
10.1229
10.1255
10.1282
10.1308
10.1334
10.1361
54
10.1387
10.1414
10.1441
10.1467
10.1494
10.1521
55°
10.1548
10.1575
10.1602
10.1629
10.1656
10.1683
56
10.1710
10.1737
10.1765
10.1792
10.1820
10.1847
57
10.1875
10.1903
10.1930
10.1958
10.1986
10.2014
58
10.2042
10.2070
10.2098
10.2127
10.2155
10.2184
59
10.2212
10.2241
10.2270
10.2299
10.2327
10.2356
60°
10.2386
10.2415
10.2444
10.2474
10.2503
10.2533
61
10.2562
10.2592
10.2622
10.2652
10.2683
10.2713
62
10.2743
10.2774
10.2804
10.2835
10.2866
10.2897
63
10.2928
10.2960
10.2991
10.3023
10.3054
10.3086
64
10.3118
10.3150
10.3183
10.3215
10.3248
10.3280
65°
10.3313
10.3346
10.3380
10.3413
10.3447
10.3480
66
10.3514
10.3548
10.3583
10.3617
10.3652
10.3686
67
10.3721
10.3757
10.3792
10.3828
10.3864
10.3900
68
10.3936
10.3972
10.4009
10.4046
10.4083
10.4121
69
10.4158
10.4196
10.4234
10.4273
10.4311
10.4350
70°
10.4389
10.4429
10.4469
10.4509
10.4549
10.4589
71
10.4630
10.4671
10.4713
10.4755
10.4797
10.4839
72
10.4g82
10.4925
10.4969
10.5013
10.5057
10.5102
73
10.5147
10.5192
10.5238
10.5284
10.5331
10.5378
74
10.5425
10.5473
10.5521
10.5570
10.5619
10.5669
15°
10.5719
10.5770
10.5822
10.5873
10.5926
10.5979
76
10.6032
10.6086
10.6141
10.6196
10.6252
10.6309
77
10.6366
10.6424
10.6483
10.6542
10.6603
10.6664
78
10.6725
10.6788
10.6851
10.6915
10.6980
10.7047
79
10.7113
10.7181
10.7250
10.7320
10.7391
10.7464
80°
10.7537
10.7611
10.7687
10.7764
10.7842
10.7922
81
10.8003
10.8085
10.8169
10.8255
10.8342
10.8431
82
10.8522
10.8615
10.8709
10.8806
10.8904
10.9005
83
10.9109
10.9214
10.9322
10.9433
10.9547
10.9664
84
10.9784
10.9907
11.0034
11.0164
11.0299
11.0437
85°
11.0580
11.0728
11.0882
11.1040
11.1205
11.1376
86
11.1554
11.1739
11.1933
11.2135
11.2348
11.2571
87
11.2806
11.3055
11.3318
11.3599
11.3899
11.4221
88
11.4569
11.4947
11.5362
11.5819
11.6331
11.6911
89
11.7581
11.8373
11.9342
12.0591
12.2352
12.5363
90°
221
TABLE
13
CONVERSrON OF RADIANS TO DEGREES,
MINUTES AND SECONDS OR FRACTIONS OF DEGREES
Radians
Deg.
Min.
Sec.
Fractions of
Degrees
1
57°
17'
44.8"
57.2958°
2
114°
35'
29.6"
114.5916°
3
171°
53'
14.4"
171.8873°
4
229°
10'
59.2"
229.1831°
5
286°
28'
44.0"
286.4789°
6
343°
46'
28.8"
343.7747°
7
8
401°
458°
4'
21'
13.6"
58.4"
401.0705°
458.3662°
9
515°
39'
43.3"
515.6620°
10
572°
57'
28.1"
572.9578°
.1
5°
43'
46.5"
.2
11°
27'
33.0"
.3
17°
11'
19.4"
.4
.5
22°
28°
65'
38'
5.9"
52.4"
.6
.7
34°
40°
22'
6'
38.9"
25.4"
.8
45°
50'
11.8"
.9
51°
33'
58.3"
.01
.02
0°
1°
34'
8'
22.6"
45.3"
.03
.04
1°
2°
43'
17'
7.9"
30.6"
.05
.06
2°
3°
51'
26'
53.2"
15.9"
.07
.08
4°
4°
0'
35'
38.5"
1.2"
.09
5°
9'
23.8"
.001
0°
3'
26.3"
.002
.003
.004
.005
0°
0°
0°
0°
6'
10'
13'
17'
52.5"
18.8"
45.1"
11.3"
.006
0°
20'
37.6"
.007
.008
0°
0°
24'
27'
3.9"
30.1"
.009
0°
30'
56.4"
.0001
0°
0'
20.6"
.0002
.0003
0°
0°
0'
1'
41.3"
1.9"
.0004
0°
1'
22.5"
.0005
0°
1'
43.1"
.0006
.0007
0°
0°
2'
2'
3.8"
24.4"
.0008
0°
2'
45.0"
.0009
0°
3'
5.6"
222
TABLE
14
CONVERSION OF DEGREES, MINUTES
AND SECONDS TO RADIANS
Deg:rees
Radians
1^
.0174533
2°
.0349066
3°
.0523599
4°
.0698132
5°
.0872665
00
.1047198
70
.1221730
8"
.1396263
9°
.1570796
10°
.1745329
Minutes
Radians
1'
.00029089
2'
.00058178
3'
.00087266
4'
.00116355
5'
.00145444
6'
.00174533
7'
.00203622
8'
.00232711
9'
.00261800
10'
.00290888
Seconds
Radians
X"
.0000048481
2"
.0000096963
3"
.0000145444
4"
.0000193925
5"
.0000242407
6"
.0000290888
7"
.0000339370
8"
.0000387851
9"
.0000436332
10"
.0000484814
223
TABLE
15
NATURAL OR NAPIERIAN LOGARITHMS
log^ X or In X
X
6
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
.00000
.09531
.18232
.26236
.33647
.40547
.47000
.53063
.58779
.64185
.69315
.74194
.78846
.83291
.87547
.91629
.95551
.99325
1.02962
1.06471
1.09861
1.13140
1.16315
1.19392
1.22378
1.25276
1.28093
1.30833
1.33500
1.36098
1.38629
1.41099
1.43508
1.45862
1.48160
1.50408
1.52606
1.54756
1.56862
1.58924
.00995
.10436
.19062
.27003
.34359
.41211
.47623
.53649
.59333
.64710
.69813
.74669
.79299
.83725
.87963
.92028
.95935
.99695
1.03318
1.06815
1.10194
1.13462
1.16627
1.19695
1.22671
1.25562
1.28371
1.31103
1.33763
1.36354
1.38879
1.41342
1.43746
1.46094
1.48387
1.50630
1.52823
1.54969
1.57070
1.59127
.01980
.11333
.19885
.27763
.35066
.41871
.48243
.54232
.59884
.65233
.70310
.75142
.79751
.84157
.88377
.92426
.96317
1.00063
1.03674
1.07158
1.10526
1.13783
1.16938
1.19996
1.22964
1.25846
1.28647
1.31372
1.34025
1.36609
1.39128
1.41585
1.43984
1.46326
1.48614
1.50851
1.53039
1.55181
1.57277
1.59331
.02956
.12222
.20701
.28518
.35767
.42527
.48858
.54812
.60432
.65752
.70804
.75612
.80200
.84587
.88789
.92822
.96698
1.00430
1.04028
1.07500
1.10856
1.14103
1.17248
1.20297
1.23256
1.26130
1.28923
1.31641
1.34286
1.36864
1.39377
1.41828
1.44220
1.46557
1.48840
1.51072
1.53256
1.55393
1.57485
1.59534
.03922
.13103
.21511
.29267
.36464
.43178
.49470
.55389
.60977
.66269
.71295
.76081
.80648
.85015
.89200
.93216
.97078
1.00796
1.04380
1.07841
1.11186
1.14422
1.17557
1.20597
1.23547
1.26413
1.29198
1.31909
1.34547
1.37118
1.39624
1.42070
1.44456
1.46787
1.49065
1.51293
1.53471
1.55604
1.57691
1.59737
.04879
.13976
.22314
.30010
.37156
.43825
.50078
.55962
.61519
.66783
.71784
.76547
.81093
.85442
.89609
.93609
.97456
1.01160
1.04732
1.08181
1.11514
1.14740
1.17865
1.20896
1.23837
1.26695
1.29473
1.32176
1.34807
1.37372
1.39872
1.42311
1.44692
1.47018
1.49290
1.51513
1.53687
1.55814
1.57898
1.59939
.05827
.14842
.23111
.30748
.37844
.44469
.50682
.56531
.62058
.67294
.72271
.77011
.81536
.85866
.90016
.94001
.97833
1.01523
1.05082
1.08519
1.11841
1.15057
1.18173
1.21194
1.24127
1.26976
1.29746
1.32442
1.35067
1.37624
1.40118
1.42552
1.44927
1.47247
1.49515
1.51732
1.53902
1.56025
1.58104
1.60141
.06766
.15700
.23902
.31481
.38526
.45108
.51282
.57098
.62594
.67803
.72755
.77473
.81978
.86289
.90422
.94391
.98208
1.01885
1.05431
1.08856
1.12168
1.15373
1.18479
1.21491
1.24415
1.27257
1.30019
1.32708
1.35325
1.37877
1.40364
1.42792
1.45161
1.47476
1.49739
1.51951
1.54116
1.56235
1.58309
1.60342
.07696
.16551
.24686
.32208
.39204
.45742
.51879
.57661
.63127
.68310
.73237
.77932
.82418
.86710
.90826
.94779
.98582
1.02245
1.05779
1.09192
1.12493
1.15688
1.18784
1.21788
1.24703
1.27536
1.30291
1.32972
1.35584
1.38128
1.40610
1.43031
1.45395
1.47705
1.49962
1.52170
1.54330
1.56444
1.58515
1.60543
.08618
.17395
.25464
.32930
.39878
.46373
.52473
.58222
.63658
.68813
.73716
.78390
.82855
.87129
.91228
.95166
.98954
1.02604
1.06126
1.09527
1.12817
1.16002
1.19089
1.22083
1.24990
1.27815
1.30563
1.33237
1.35841
1.38379
1.40854
1.43270
1.45629
1.47933
1.50185
1.52388
1.54543
1.56653
1.58719
1.60744
In 10 = 2.30259
2 In 10 = 4.60517
3 In 10 = 6.90776
4 In 10 =
9.21034
7 In 10 ^
^ 16.11810
5 In 10 ^
11.51293
8 In 10 ^
^ 18.42068
6 In 10 =
13.81551
9 In 10 =
= 20.72327
224
Table 15
(continued)
NATURAL OR NAPIERIAN LOGARITHMS
log, X or In X
X
1
2
3
4
5
6
7
8
9
5.0
1.60944
1.61144
1.61343
1.61542
1.61741
1.61939
1.62137
1.62334
1.62531
1.62728
5.1
1.62924
1.63120
1.63315
1.63511
1.63705
1.63900
1.64094
1.64287
1.64481
1.64673
5.2
1.64866
1,65058
1.65250
1.65441
1.65632
1.65823
1.66013
1.66203
1.66393
1.66582
6.3
1.66771
1.66959
1.67147
1.67335
1.67523
1.67710
1.67896
1.68083
1.68269
1.68455
5.4
1.68640
1.68825
1.69010
1.69194
1.69378
1.69562
1.69745
1.69928
1.70111
1.70293
5.6
1.70475
1.70656
1.70838
1.71019
1.71199
1.71380
1.71560
1.71740
1.71919
1.72098
6.6
1.72277
1.72455
1.72633
1.72811
1.72988
1.73166
1.73342
1.73519
1.73695
1.73871
6.7
1.74047
1.74222
1.74397
1.74572
1.74746
1.74920
1.75094
1.75267
1.75440
1.75613
5.8
1.75786
1.75958
1.76130
1.76302
1.76473
1.76644
1.76815
1.76986
1.77156
1.77326
5.9
1.77495
1.77665
1.77834
1.78002
1.78171
1.78339
1.78507
1.78675
1.78842
1.79009
6.0
1.79176
1.79342
1.79509
1.79675
1.79840
1.80006
1.80171
1.80336
1.80500
1.80665
6.1
1.80829
1.80993
1.81156
1.81319
1.81482
1.81645
1.81808
1.81970
1.82132
1.82294
6.2
1.82455
1.82616
1.82777
1.82938
1.83098
1.83258
1.8.3418
1.83578
1.83737
1.83896
6.3
1.84055
1.84214
1.84372
1.84530
1.84688
1.84845
1.85003
1.85160
1.85317
1.85473
6.4
1.85630
1.85786
1.85942
1.86097
1.86253
1.86408
1.86563
1.86718
1.86872
1.87026
6.5
1.87180
1.87334
1.87487
1.87641
1.87794
1.87947
1.88099
1.88251
1.88403
1.88555
6.6
1.88707
1.88858
1.89010
1.89160
1.89311
1.89462
1.89612
1.89762
1.89912
1.90061
6.7
1.90211
1.90360
1.90509
1.90658
1.90806
1.90954
1.91102
1.91250
1.91398
1.91545
6.8
1.91692
1.91839
1.91986
1.92132
1.92279
1.92425
1.92571
1.92716
1.92862
1.93007
6.9
1.93152
1.93297
1.93442
1.93586
1.93730
1.93874
1.94018
1.94162
1.94306
1.94448
7.0
1.94591
1.94734
1.94876
1.95019
1.95161
1.95303
1.95445
1.95586
1.95727
1.95869
7.1
1.96009
1.96150
1.96291
1.96431
1.96571
1.96711
1.96851
1.96991
1.97130
1.97269
7.2
1.97408
1.97547
1.97685
1.97824
1.97962
1.98100
1.98238
1.98376
1.98513
1.98650
7.3
1.98787
1.98924
1.99061
1.99198
1.99334
1.99470
1.99606
1.99742
1.99877
2.00013
7.4
2.00148
2.00283
2.00418
2.00553
2.00687
2.00821
2.00956
2.01089
2.01223
2.01357
7.5
2.01490
2.01624
2.01757
2.01890
2.02022
2.02155
2.02287
2.02419
2.02551
2.02683
7.6
2.02815
2.02946
2.03078
2.03209
2.03340
2.03471
2.03601
2.03732
2.03862
2.03992
7.7
2.04122
2.04252
2.04381
2.04511
2.04640
2.04769
2.04898
2.05027
2.06156
2.05284
7.8
2.05412
2.05540
2.05668
2.05796
2.05924
2.06051
2.06179
2.06306
2.06433
2.06560
7.9
2.06686
2.06813
2.06939
2.07065
2.07191
2.07317
2.07443
2.07568
2.07694
2.07819
8.0
2.07944
2.08069
2.08194
2.08318
2.08443
2.08567
2.08691
2.08815
2.08939
2.09063
8.1
2.09186
2.09310
2.09433
2.09556
2.09679
2.09802
2.09924
2.10047
2.10169
2.10291
8.2
2.10413
2.10535
2.10657
2.10779
2.109O0
2.11021
2.11142
2.11263
2.11384
2.11505
8.3
2.11626
2.11746
2.11866
2.11986
2.12106
2.12226
2.12346
2.12466
2.12585
2.12704
8.4
2.12823
2.12942
2.13061
2.13180
2.13298
2.13417
2.13535
2.13653
2.13771
2.13889
8.5
2.14007
2.14124
2.14242
2.14359
2.14476
2.14593
2.14710
2.14827
2.14943
2.15060
8.6
2.15176
2.15292
2.15409
2.15524
2.15640
2.15756
2.15871
2.15987
2.16102
2.16217
8.7
2.16332
2.16447
2.16562
2.16677
2.16791
2.16906
2.17020
2.17134
2.17248
2.17361
8.8
2.17475
2.17589
2.17702
2.17816
2.17929
2.18042
2.18155
2.18267
2.18380
2.18493
8.9
2.18605
2.18717
2.18830
2.18942
2.19054
2.19165
2.19277
2.19389
2.19500
2.19611
9.0
2.19722
2.19834
2.19944
2.20055
2.20166
2.20276
2.20387
2.20497
2.20607
2.20717
9.1
2.20827
2.20937
2.21047
2.21157
2.21266
2.21375
2.21485
2.21594
2.21703
2.21812
9.2
2.21920
2.22029
2.22138
2.22246
2.22354
2.22462
2.22570
2.22678
2.22786
2.22894
9.3
2.23001
2.23109
2.23216
2.23324
2.23431
2.23538
2.23645
2.23761
2.23858
2.23965
9.4
2'.24071
2.24177
2.24284
2.24390
2.24496
2.24601
2.24707
2.24813
2.24918
2.25024
9.5
2.25129
2.25234
2.25339
2.25444
2.25549
2.25654
2.25759
2.25863
2.25968
2.26072
9.6
2.26176
2.26280
2.26384
2.26488
2.26592
2.26696
2.26799
2.26903
2.27006
2.27109
9.7
2.27213
2.27316
2.27419
2.27521
2.27624
2.27727
2.27829
2.27932
2.28034
2.28136
9.8
2.28238
2.28340
2.28442
2.28544
2.28646
2.28747
2.28849
2.28950
2.29051
2.29152
9.9
2.29253
2.29354
2.29455
2.29556
2.29657
2.29757
2.29858
2.29958
2.30068
2.30158
225
X
1
2
3
4
5
6
7
8
9
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1.0000
1.0101
1.0202
1.0305
1.0408
1.0513
1.0618
1.0725
1.0833
1.0942
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1.1052
1.1163
1.1275
1.1388
1.1503
1.1618
1.1735
1.1853
1.1972
1.2092
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1.2214
1.2337
1.2461
1.2586
1.2712
1.2840
1.2969
1.3100
1.3231
1.3364
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1.3499
1.3634
1.3771
1.3910
1.4049
1.4191
1.4333
1.4477
1.4623
1.4770
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1.4918
1.5068
1.5220
1.5373
1.5527
1.5683
1.5841
1.6000
1.6161
1.6323
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1.6487
1.6653
1.6820
1.6989
1.7160
1.7333
1.7507
1.7683
1.7860
1.8040
.6
1.8221
1.8404
1.8589
1.8776
1.8965
1.9155
1.9348
1.9542
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1.9937
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2.0138
2.0340
2.0544
2.0751
2.0959
2.1170
2.1383
2.1598
2.1815
2.2034
.8
2.2255
2.2479
2.2705
2.2933
2.3164
2.3396
2.3632
2.38G9
2.4109
2.4351
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2.4596
2.4843
2.5093
2.5345
2.5600
2.5857
2,6117
2.6379
2.6645
2.6912
1.0
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2.7456
2.7732
2.8011
2.8292
2.8577
2.8864
2,9154
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2.9743
1.1
3.0042
3.0344
3.0649
3.0957
3.1268
3.1582
3.1899
3.2220
3.2544
3.2871
1.2
3.3201
3.3535
3.3872
3.4212
3.4556
3.4903
3.5254
3.5609
3.5966
3.6328
1.3
3.6693
3.7062
3.7434
3.7810
3.8190
3.8574
3.8962
3.9354
3.9749
4.0149
1.4
4.0552
4.0960
4.1371
4.1787
4.2207
4.2631
4.3060
4.3492
4.3929
4.4371
1.5
4.4817
4.5267
4.5722
4.6182
4.6646
4.7115
4.7588
4.8066
4.8550
4.9037
1.6
4.9530
5.0028
5.0531
5.1039
5.1552
5.2070
5.2593
5.3122
5.3656
5.4195
1.7
5.4739
5.5290
5.5845
5.6407
5.6973
5.7546
5.8124
5.8709
5.9299
5.9895
1.8
6.0496
6.1104
6.1719
6.2339
6.2965
6.3598
6.4237
6.4883
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6.6194
1.9
6.6859
6.7531
6.8210
6.8895
6.9588
7.0287
7.0993
7.1707
7.2427
7.3155
2.0
7.3891
7.4633
7.5383
7.6141
7.6906
7.7679
7.8460
7.9248
8.0045
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8.1662
8.2482
8.3311
8.4149
8.4994
8.5849
8.6711
8.7583
8.8463
8.9352
2.2
9.0250
9.1157
9.2073
9.2999
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9.4877
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2.3
9.9742
10.074
10.176
10.278
10.381
10.486
10.591
10.697
10.805
10.913
2.4
11.023
11.134
11.246
11.359
11.473
11.588
11.705
11.822
11.941
12.061
2.5
12.182
12.305
12.429
12.554
12.680
12.807
12.936
13.066
13.197
13.330
2.6
13.464
13.599
13.73G
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14.013
14.154
14.296
14.440
14.585
14.732
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14.880
15.029
15.180
15.333
15.487
15.643
15.800
15.959
16.119
16.281
2.8
16.445
16.610
16.777
16.945
17.116
17.288
17.462
17.637
17.814
17.993
2.9
18.174
18.357
18.541
18.728
18.916
19.106
19.298
19.492
19.688
19.886
3.0
20.086
20.287
20.491
20.697
20.905
21.115
21.328
21.542
21.758
21.977
3.1
22.198
22.421
22.646
22.874
23.104
23.336
23.571
23.807
24.047
24.288
3.2
24.533
24.779
25.028
25.280
25.534
25.790
26.050
26.311
26.576
26.843
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27.113
27.385
27.660
27.938
28.219
28.503
28.789
29.079
29.371
29.666
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29.964
30.265
30.569
30.877
31.187
31.500
31.817
32.137
32.460
32.786
3.5
33.115
33.448
33.784
34.124
34.467
34.813
35.163
35.517
35.874
36.234
3.6
36.598
36.966
37.338
37.713
38.092
38.475
38.861
39.252
39.646
40.045
3.7
40.447
40.854
41.264
41.679
42.098
42.521
42.948
43.380
43.816
44.256
3.8
44.701
45.150
45.604
46.063
46.525
46.993
47.465
47.942
48.424
48.911
3.9
49.402
49.899
50.400
50.907
51.419
51.935
52.457
52.985
53.517
54.055
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54.598
60.340
66.686
73.700
81.451
90.017
99.484
109.95
121.51
134.29
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148.41
164.02
181.27
200.34
221.41
244.69
270.43
298.87
330.30
365.04
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403.43
445.86
492.75
544.57
601.85
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812.41
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5
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1.00000
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1.5
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1.6
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1.7
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1.8
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1.9
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2.0
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2.2
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2.5
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2.1059
1.5
2.1293
2.1529
2.1768
2.2008
2.2251
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2.2743
2.2993
2.3245
2.3499
1.6
2.3756
2.4015
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2.4806
2.5075
2.5346
2.5620
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2.6175
1.7
2.6456
2.6740
2.7027
2.7317
2.7609
2.7904
2.8202
2.8503
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2.9112
1.8
2.9422
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3.0049
3.0367
3.0689
3.1013
3.1340
3.1671
3.2005
3.2341
1.9
3.2682
3.3025
3.3372
3.3722
3.4075
3.4432
3.4792
3.5156
3.5523
3.5894
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3.6269
3.6647
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3.7414
3.7803
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3.8593
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4.0219
4.0635
4.1056
4.1480
4.1909
4.2342
4.2779
4.3221
4.3666
4.4116
2.2
4.4571
4.5030
4.5494
4.5962
4.6434
4.6912
4.7394
4.7880
4.8372
4.8868
2.3
4.9370
4.9876
5.0387
5.0903
5.1425
5.1951
5.2483
5.3020
5.3562
5.4109
2.4
5.4662
5.5221
5.5785
5.6354
5.6929
5.7510
5.8097
5.8689
5.9288
5.9892
2.5
6,0502
6.1118
6.1741
6.2369
6.3004
6.3645
6.4293
6.4946
6.5607
6.6274
2.6
6.6947
6.7628
6.8315
6.9008
6.9709
7.0417
7.1132
7.1854
7.2583
7.3319
2.7
7.4063
7.4814
7.5572
7.6338
7.7112
7.7894
7.8683
7.9480
8.0285
8.1098
2.8
8.1919
8.2749
8.3586
8.4432
8.5287
8.6150
8.7021
8.7902
8.8791
8.9689
2.9
9.0596
9.1512
9.2437
9.3371
9.4315
9.5268
9.6231
9.7203
9.8185
9.9177
228
Table 18a
(continued)
HYPERBOLIC FUNCTIONS
sinh X
X
1
2
3
4
5
6
7
8
9
3.0
10.018
10.119
10.221
10.324
10.429
10.534
10.640
10.748
10.856
10.966
3.1
11.076
11.188
11.301
11.415
11.530
11.647
11.764
11.883
12.003
12.124
3.2
12.246
12.369
12.494
12.620
12.747
12.876
13.006
13.137
13.269
13.403
3.3
13.538
13.674
13.812
13.951
14.092
14.234
14.377
14.522
14.668
14.816
3.4
14.965
15.116
15.268
15.422
15.577
15.734
15.893
16.053
16.215
16.378
3.5
16.543
16.709
16.877
17.047
17.219
17.392
17.567
17.744
17.923
18.103
3.6
18.285
18.470
18.655
18.843
19.033
19.224
19.418
19.613
19.811
20.010
3.7
20.211
20.415
20.620
20.828
21.037
21.249
21.463
21.679
21.897
22.117
3.8
22.339
22.564
22.791
23.020
23.252
23.486
23.722
23.961
24.202
24.445
3.9
24.691
24.939
25.190
25.444
25.700
25.958
26.219
26.483
26.749
27.018
4.0
27.290
27.564
27.842
28.122
28.404
28.690
28.979
29.270
29.564
29.862
4.1
30.162
30.465
30.772
31.081
31.393
31.709
32.028
32.350
32.675
33.004
4.2
33.336
33.671
34.009
34.351
34.697
35.046
35.398
35.754
36.113
36.476
4.3
36.843
37.214
37.588
37.965
38.347
38.733
39.122
39.515
39.913
40.314
4.4
40.719
41.129
41.542
41.960
42.382
42.808
43.238
43.673
44.112
44.555
4.5
45.003
45.455
45.912
46.374
46.840
47.311
47.787
48.267
48.752
49.242
4.6
49.737
60.237
50.742
51.252
51.767
52.288
52.813
63.344
53.880
54.422
4.7
54.969
55.522
56.080
56.643
57.213
57.788
58.369
58.995
59.548
60.147
4.8
60.751
61.362
61.979
62.601
63.231
63.866
64.508
65.157
65.812
66.473
4.9
67.141
67.816
68.498
69.186
69.882
70.584
71.293
72.010
72.734
73.465
5.0
74.203
74.949
75.702
76.463
77.232
78.008
78.792
79.584
80.384
81.192
5.x
82.008
82.832
83.665
84.506
85.355
86.213
87.079
87.955
88.839
89.732
5.2
90.633
91.544
92.464
93.394
94.332
95.281
96.238
97.205
98.182
99.169
5.3
100.17
101.17
102.19
103.22
104.25
105.30
106.36
107.43
108.51
109.60
5.4
110.70
111.81
112.94
114.07
115.22
116.38
117.55
118.73
119.92
121.13
5.5
122.34
123.57
124.82
126.07
127.34
128.62
129.91
131.22
132.53
133.87
5.6
135.21
136.57
137.94
139.33
140.73
142.14
143.57
145.02
146.47
147.95
5.7
149.43
150.93
152.45
153.98
155.53
157.09
158.67
160.27
161.88
163.51
5.8
165.15
166.81
168.48
170.18
171.89
173.62
175.36
177.12
178.90
180.70
5.9
182.52
184.35
186.20
188.08
189.97
191.88
193.80
195.75
197.72
199.71
229
X
1
2
3
4
5
6
7
8
9
.0
1.0000
1.0001
1.0002
1.0005
1.0008
1.0013
1.0018
1.0025
1.0032
1.0041
.1
1.0050
1.0061
1.0072
1.0085
1.0098
1.0113
1.0128
1.0145
1.0162
1.0181
.2
1.0201
1.0221
1.0243
1.0266
1.0289
1.0314
1.0340
1.0367
1.0395
1.0423
.3
1.0453
1.0484
1.0516
1.0549
1.0584
1.0619
1.0655
1,0692
1.0731
1.0770
A
1.0811
1.0852
1.0895
1.0939
1.0984
1.1030
1.1077
1,1125
1.1174
1.1225
.5
1.1276
1.1329
1.1383
1.1438
1,1494
1.1551
1.1609
1.1669
1.1730
1.1792
.6
1.1855
1.1919
1.1984
1.2051
1.2119
1.2188
1.2258
1.2330
1.2402
1.2476
.7
1.2552
1.2628
1.2706
1.2785
1.2865
1.2947
1.3030
1.3114
1.3199
1.3286
.8
1.3374
1.3464
1.3555
1.3647
1.3740
1.3835
1.3932
1.4029
1.4128
1.4229
.9
1.4331
1.4434
1.4539
1.4645
1.4753
1.4862
1.4973
1.5085
1.5199
1.5314
1.0
1.5431
1.5549
1.5669
1.5790
1.5913
1.6038
1.6164
1.6292
1.6421
1.6552
1.1
1.6685
1.6820
1.6956
1.7093
1.7233
1.7374
1.7517
1.7662
1.7808
1.7957
1.2
1.8107
1.S258
1.8412
1.8568
1.8725
1.8884
1.9045
1.9208
1.9373
1.9540
1.3
1.9709
1.9880
2.0053
2.0228
2.0404
2.0583
2.0764
2.0947
2.1132
2.1320
1.4
2.1509
2.1700
2.1894
2.2090
2.2288
2.2488
2.2691
2.2896
2.3103
2.3312
1.5
2.3524
2.3738
2.3955
2.4174
2.4395
2.4619
2.4845
2.5073
2.5305
2.5538
1.6
2.5775
2.6013
2.6255
2.6499
2.6746
2.6995
2.7247
2.7502
2.7760
2.8020
1.7
2.8283
2.8549
2.8818
2.9090
2.9364
2.9642
2.9922
3.0206
3.0492
3.0782
1.8
3.1075
3.1371
3.1669
3.1972
3.2277
3.2585
3.2897
3.3212
3.3530
3.3852
1.9
3.4177
3.4506
3.4838
3.5173
3.5512
3.5855
3.6201
3.6551
3.6904
3.7261
2.0
3.7622
3.7987
3.8355
3.8727
3.9103
3.9483
3.9867
4,0255
4.0647
4.1043
2.1
4.1443
4.1847
4.2256
4.2669
4.3085
4.3507
4.3932
4.4362
4.4797
4.5236
2.2
4.5679
4.6127
4.6580
4.7037
4.7499
4.7966
4.8437
4.8914
4.9395
4.9881
2.3
5.0372
5.0868
5.1370
5.1876
5.2388
5.2905
5.3427
5.3954
5.4487
5.5026
2.4
5.5569
5.6119
5.6674
6.7235
5.7801
5.8373
5.8951
5,9535
6.0125
6.0721
2.5
6.1323
6.1931
6.2545
6.3166
6.3793
6.4426
6.5066
6.5712
6.6365
6.7024
2.6
6.7690
6.8363
6.9043
6.9729
7.0423
7.1123
7.1831
7.2546
7.3268
7.3998
2.7
7.4735
7.5479
7.6231
7.6991
7.7758
7.8533
7.9316
8.0106
8.0905
8.1712
2.8
8.2527
8.3351
8.4182
8.5022
8.5871
8.6728
8.7594
8.8469
8.9352
9.0244
2.9
9.1146
9.2056
9.2976
9.3905
9.4844
9.5791
9.6749
9.7716
9.8693
9.9680
230
Table 18b
(con tinned)
HYPERBOLIC FUNCTIONS
cosh X
X
1
2
3
4
5
6
7
8
9
3.0
10.068
10.168
10.270
10.373
10.476
10.581
10.687
10.794
10.902
11.011
3.1
11.121
11.233
11.345
11.459
11.574
11.689
11.806
11.925
12.044
12.165
3.2
12.287
12.410
12.534
12.660
12.786
12.915
13.044
13.175
13.307
13.440
3.3
13.575
13.711
13.848
13.987
14.127
14.269
14.412
14.556
14.702
14.850
3.4
14.999
15.149
15.301
15.455
15.610
15.766
15.924
16.084
16.245
16.408
3.5
16.573
16.739
16.907
17.077
17.248
17.421
17.596
17.772
17.951
18.131
3.6
18.313
18.497
18.682
18.870
19.059
19.250
19.444
19.639
19.836
20.035
3.7
20.236
20.439
20.644
20.852
21.061
21.272
21.486
21.702
21.919
22.139
3.8
22.302
22.586
22.813
23.042
23.273
23.507
23.743
23.982
24.222
24.466
3.9
24.711
24.959
25.210
25.463
25.719
25.977
26.238
26.502
26.768
27.037
4.0
27.308
27.583
27.860
28.139
28.422
28.707
28.996
29.287
29.581
29.878
4.1
30.178
30.482
30.788
31.097
31.409
31.725
32.044
32.365
32.691
33.019
4.2
33.351
33.686
34.024
34.366
34.711
35.060
35.412
35.768
36.127
36.490
4.3
36.857
37.227
37.601
37.979
38.360
38.746
39.135
39.528
39.925
40.326
4.4
40.732
41.141
41.554
41.972
42.393
42.819
43.250
43.684
44.123
44.566
4.5
45.014
45.466
45.923
46.385
46.851
47.321
47.797
48.277
48.762
49.252
4.6
49.747
50.247
50.752
51.262
51.777
52.297
52.823
53.354
53.890
54.431
4.7
54.978
55.531
66.089
56.652
57.221
57.796
58.377
58.964
59.556
60.155
4.8
60.759
61.370
61.987
62.609
63.239
63.874
64.516
65.164
65.819
66.481
4.9
67.149
67.823
68.505
69.193
69.889
70.591
71.300
72.017
72.741
73.472
5.0
74.210
74.956
75.709
76.470
77.238
78.014
78.798
79.590
80.390
81.198
5.1
82.014
82.838
83.671
84.512
85.361
86.219
87.085
87.960
88.844
89.737
5.2
90.639
91.550
92.470
93.399
94.338
95.286
96.243
97.211
98.188
99.174
5.3
100.17
101.18
102.19
103.22
104.26
105.31
106.67
107.43
108.51
109.60
5.4
110.71
111.82
112.94
114.08
115.22
116.38
117.55
118.73
119.93
121.13
5.5
122.35
123.58
124.82
126.07
127.34
128.62
129.91
131.22
132.54
133.87
5.6
135.22
136.57
137.95
139.33
140.73
142.15
143.58
145.02
146.48
147.95
5.7
149.44
150.94
152.45
153.99
155.53
157.10
158.68
160.27
161.88
163.51
5.8
165.15
166.81
168.49
170.18
171.89
173.62
175.36
177.13
178.91
180.70
5.9
182.52
184.35
186.21
188.08
189.97
191.88
193.81
195.75
197.72
199.71
231
X
1
2
3
4
5
6
7
8
9
.0
.00000
.01000
.02000
.02999
.03998
.04996
.05993
.06989
.07983
.08976
.1
.09967
.10956
.11943
.12927
.13909
.14889
.15865
.16838
.17808
.18775
.2
.19738
.20697
.21652
.22603
.23550
.24492
.25430
.26362
.27291
.28213
.3
.29131
.30044
.30951
.31852
.32748
.33638
.34521
.35399
.36271
.37136
.4
.37995
.38847
.39693
.40532
.41364
.42190
.43008
.43820
.44624
.45422
.5
.46212
.46995
.47770
.48538
.49299
.50052
.50798
.51536
.52267
.52990
.6
.53705
.54413
.55113
.55805
.56490
.57167
.57836
.58498
.59152
.59798
.7
.60437
.61068
.61691
.62307
.62915
.63515
.64108
.64693
.65271
.65841
.8
.66404
.66959
.67507
.68048
.68581
.69107
.69626
.70137
.70642
.71139
.9
.71630
.72113
.72590
.73059
.73522
.73978
.74428
.74870
.75307
.75736
1.0
.76159
.76576
.76987
.77391
.77789
.78181
.78566
.78946
.79320
.79688
1.1
.80050
.80406
.80757
.81102
.81441
.81775
.82104
.82427
.82745
.83058
1.2
.83365
.83668
.83965
.84258
.84546
.84828
.85106
.85380
.85648
.85913
1.3
.86172
.86428
.86678
.86925
.87167
.87405
.87639
.87869
.88095
.88317
1.4
.88535
.88749
.88960
.89167
.89370
.89569
.89765
.89958
.90147
.90332
1.5
.90515
.90694
.90870
.91042
.91212
.91379
.91542
.91703
.91860
.92015
1.6
.92167
.92316
.92462
.92606
.92747
.92886
.93022
.93155
.93286
.93415
1.7
.93541
.93665
.93786
.93906
.94023
.94138
.94250
.94361
.94470
.94576
1.8
.94681
.94783
.94884
.94983
.95080
.95175
.95268
.95359
.95449
.95537
1.9
.95624
.95709
.95792
.95873
.95953
.96032
.96109
.96185
.96259
.96331
2.0
.96403
.96473
.96541
.96609
.96675
.96740
.96803
.96865
.96926
.96986
2.1
.97045
.97103
.97159
.97215
.97269
.97323
.97375
.97426
.97477
.97526
2.2
.97574
.97622
.97668
.97714
.97759
.97803
.97846
.97888
.97929
.97970
2.3
.98010
.98049
.98087
.98124
.98161
.98197
.98233
.98267
.98301
.98335
2.4
.98367
.98400
.98431
.98462
.98492
.98522
.98551
.98579
.98607
.98635
2.5
.98661
.98688
.98714
.98739
.98764
.98788
.98812
.98835
.98858
.98881
2.6
.98903
.98924
.98946
.98966
.98987
.99007
.99026
.99045
.99064
.99083
2.7
.99101
.99118
.99136
.99153
.99170
.99186
.99202
.99218
.99233
.99248
2.8
.99263
.99278
.99292
.99306
.99320
.99333
.99346
.99359
.99372
.99384
2.9
.99396
.99408
.99420
.99431
.99443
.99454
.99464
.99475
.99485
.99496
232
Table 18c
(continued)
HYPERBOLIC FUNCTIONS
tanh X
X
1
2
3
4
5
6
7
8
9
3.0
.99505
.99515
.99525
.99534
.99543
.99552
.99561
.99570
.99578
.99587
3.1
.99595
.99603
.99611
.99618
.99626
.99633
.99641
.99648
.99655
.99662
3.2
.99668
.99675
.99681
.99688
.99694
.99700
.99706
.99712
.99717
.99723
3.3
.99728
.99734
.99739
.99744
.99749
.99754
.99759
.99764
.99768
.99773
3.4
.99777
.99782
.99786
.99790
.99795
.99799
.99803
.99807
.99810
.99814
3.5
.99818
.99821
.99825
.99828
.99832
.99835
.99838
.99842
.99845
.99848
3.6
.99851
.99853
.99857
.99859
.99862
.99865
.99868
.99870
.99873
.99875
3.7
.99878
.99880
.99883
.99885
.99887
.99889
.99892
.99894
.99896
.99898
3.8
.99900
.99902
.99904
.99906
.99908
.99909
.99911
.99913
.99915
.99916
3.9
.99918
.99920
.99921
.99923
.99924
.99926
.99927
.99929
.99930
.99932
4.0
.99933
.99934
.99936
.99937
.99938
.99939
.99941
.99942
.99943
.99944
4.1
.99945
.99946
.99947
.99948
.99949
.99950
.99951
.99952
.99953
.99954
4.2
.99955
.99956
.99957
.99958
.99958
.99959
.99960
.99961
.99962
.99962
4.3
.99963
.99964
.99965
.99966
.99966
.99967
.99967
.99968
.99969
.99969
4.4
.99970
.99970
.99971
.99972
.99972
.99973
.99973
.99974
.99974
.99975
4.5
.99975
.99976
.99976
.99977
.99977
.99978
.99978
.99979
.99979
.99979
4.6
.99980
.99980
.99981
.99981
.99981
.99982
.99982
.99982
.99983
.99983
4.7
.99983
.99984
.99984
.99984
.99985
.99985
.99985
.99986
.99986
.99986
4.8
.99986
.99987
.99987
.99987
.99987
.99988
.99988
.99988
.99988
.99989
4.9
.99989
.99989
.99990
.99990
.99990
.99990
.99990
.99990
.99991
.99991
5.0
.99991
.99991
.99991
.99991
.99992
.99992
.99992
.99992
.99992
.99992
5.1
.99993
.99993
.99993
.99993
.99993
.99993
.99993
.99994
.99994
.99994
5.2
.99994
.99994
.99994
.99994
.99994
.99994
.99995
.99995
.99995
.99995
5.3
.99995
.99995
.99995
.99995
.99995
.99995
.99996
.99996
.99996
.99996
5.4
.99996
.99996
.99996
.99996
.99996
.99996
.99996
.99996
.99997
.99997
5.5
.99997
.99997
.99997
.99997
.99997
.99997
.99997
.99997
.99997
.99997
5.6
.99997
.99997
.99997
.99997
.99997
.99998
.99998
.99998
.99998
.99998
5.7
.99998
.99998
.99998
.99998
.99998
.99998
.99998
.99998
.99998
.99998
5.8
.99998
.99998
.99998
.99998
.99998
.99998
.99998
.99998
.99998
.99998
5.9
.99998
.99999
.99999
.99999
.99999
.99999
.99999
.99999
.99999
.99999
233
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
nl
1 (by definition)
1
2
6
24
120
720
5040
40,320
362,880
3,628,800
39,916,800
479,001,600
6,227,020,800
87,178,291,200
1,307,674,368,000
20,922,789,888,000
355,687,428,096,000
6,402,373,705,728,000
121,645,100,408,832,000
2,432,902,008,176,640,000
51,090,942,171,709,440,000
1,124,000,727,777,937,680,000
25,852,016,738,892,566,840,000
620,448,401,733,421,599,360,000
15,511,210,043,336,539,984,000,000
403,291,461,126,724,039,584,000,000
10,888,869,450,421,549,068,768,000,000
304,888,344,611,803,373,925,504,000,000
8,841,761,993,742,297,843,839,616,000,000
265,252,859,812,268,935,315,188,480,000,000
8.22284 X 1033
2.63131 X 1035
8.68332 X 1036
2.95233 X 1038
1.03331 X 10-to
3.71993 X 10^1
1.37638 X 10«
5.23023 X 10^"
2.03979 X 1046
n
»!
n
n\
40
8.15915 X 10-1^
80
7.15695X10118
41
3.34525 X lO-")
81
5.79713 X 10120
42
1.40501 XlO^'
82
4.75364 X 10122
43
6.04153 X 10S2
83
3.94552 X 10124
44
2.65827 X lO^-i
84
3.31424 X 10128
45
1.19622 X 1056
85
2.81710X10128
46
5.50262 X 1057
86
2.42271 X 10130
47
2.58623 X 1059
87
2.10776 X 10132
48
1.24139 X 1061
88
1.85483 X 1013-1
49
6.08282 X 1062
89
1.65080 X 10138
50
3.04141 X 1061
90
1.48572 X 10138
51
1.55112 X 1066
91
1.35200 X 10140
52
8.06582 X 10^7
92
1.24384 X 101*2
53
4.27488 X 1069
93
1.15677 X 10144
54
2.30844 X lO'i
94
1.08737 X 10i«
55
1.26964 X 10^3
95
1.03300 X 10148
56
7.10999 X 1074
96
9.91678 X 10149
57
4.05269 X 10"6
97
9.61928 X 10151
58
2.35056 X 10^8
98
9.42689 X 10153
59
1.38683 X 1080
99
9.33262 X 10155
60
8.32099 X 10^1
100
9.33262 X 10157
61
5.07580 X 1083
62
3.14700 X 1085
63
1.98261 X 1087
64
1.26887 X 1083
65
8.24765 X 1090
66
5.44345 X 1092
67
3.64711X1091
68
2.48004 X 1096
69
1.71122 X 1098
70
1.19786 X lO'Of
71
8.50479 X lOioi
72
6.12345 X 10103
73
4.47012 X 10105
74
3.30789 X 10107
75
2.48091 X 10109
76
1.88549 X 101"
77
1.45183 X 10113
78
1.13243 X 10115
79
8.94618 X 10116
234
TABLE
20
GAMMA FUNCTION
T{x) = f i^-^e-*di for 1^ a; ^2
[For other values use the formula r(a: + 1) = x r(x)]
X
r{x)
1.00
1.00000
1.01
.99433
1.02
.98884
1.03
.98355
1.04
.97844
1.05
.97350
1.06
.96874
1.07
.96415
1.08
.95973
1.09
.95546
1.10
.95135
1.11
.94740
1.12
.94359
1.13
.93993
1.14
.93642
1.15
.93304
1.16
.92980
1.17
.92670
1.18
.92373
1.19
.92089
1.20
.91817
1.21
.91558
1.22
.91311
1.23
.91075
1.24
.90852
1.26
.90640
1.26
.90440
1.27
.90250
1.28
.90072
1.29
.89904
1.30
.89747
1.31
.89600
1.82
.89464
1.38
.89338
1.34
.89222
t.36
.89115
1.86
.89018
1.S7
.88931
1.88
.88854
1.39
.88785
1.40
.88726
IM
.88676
1.42
.88636
1.43
.88604
1.44
.88581
1.46
.88566
1.46
.88560
1.47
.88563
1.48
.88575
1.49
.88595
1.50
.88623
X
r(x)
1.50
.88623
1.51
.88659
1.52
.88704
1.53
.88757
1.54
.88818
1.55
.88887
1.56
.88964
1.57
.89049
1.58
.89142
1.59
.89243
1.60
.89352
1.61
.89468
1.62
.89592
1.63
.89724
1.64
.89864
1.65
.90012
1.66
.90167
1.67
.90330
1.68
.90500
1.69
.90678
1.70
.90864
1.71
.91057
1.72
.91258
1.73
.91467
1.74
.91683
1.75
.91906
1.76
.92137
1.77
.92376
1.78
.92623
1.79
.92877
1.80
.93138
1.81
.93408
1.82
.93685
1.83
.93969
1.84
.94261
1.85
.94561
1.86
.94869
1.87
.95184
1.88
.95507
1.89
.95838
1.90
.96177
1.91
.96523
1.92
.96877
1.93
.97240
1.94
.97610
1.95
.97988
1.96
.98374
1.97
.98768
1.98
.99171
1.99
.99581
2.00
1.00000
235
TABLE
21
BINOMIAL COEFFICIENTS
w! _ n{n -!)■ • ■jn-k + l) _
klin-k)\ ~ kl
n
n~k
0!=1
Note that each number is the sum of two numbers in the row above; one of these numbers is in the same
column and the other is in the preceding column [e.g. 56 - 35 + 21]. The arrangement is often called
Pascal's triangle [see 3.6, page 41.
n ^\
1
2
3
4
5
6
7
8
9
1
1 1
2
1 2
1
3
1 3
3
1
4
1 4
6
4
1
5
1 5
10
10
5
1
6
1 6
15
20
15
6
1
7
1 7
21
35
35
21
7
1
8
1 8
28
56
70
56
28
8
1
9
1 9
36
84
126
126
84
36
9
1
10
1 10
45
120
210
252
210
120
45
10
11
1 11
56
165
330
462
462
330
166
55
12
1 12
66
220
495
792
924
792
495
220
13
1 13
78
286
715
1287
1716
1716
1287
715
14
1 14
91
364
1001
2002
3003
3432
3003
2002
15
1 15
105
456
1365
3003
5005
6435
6435
5005
16
1 16
120
560
1820
4368
8008
11440
12870
11440
17
1 17
136
680
2380
6188
12376
19448
24310
24310
18
1 18
153
816
3060
8568
18564
31824
43758
48620
19
1 19
171
969
3876
11628
27132
50388
75582
92378
20
1 20
190
1140
4845
15504
38760
77620
125970
167960
21
1 21
210
1330
5985
20349
54264
116280
203490
293930
22
1 22
231
1540
7315
26334
74613
170544
319770
497420
23
1 23
253
1771
8855
33649
100947
245157
490314
817190
24
1 24
276
2024
10626
42504
134596
346104
735471
1307504
25
1 25
300
2300
12650
53130
177100
480700
1081575
2042975
26
1 26
325
2600
14950
65780
230230
657800
1562275
3124550
27
1 27
351
2925
17550
80730
296010
888030
2220075
4686825
28
1 28
378
3276
20475
98280
376740
1184040
3108105
6906900
29
1 29
406
3654
23751
118755
475020
1560780
4292145
10015005
30
1 30
435
4060
27405
142506
593775
2035800
5852925
14307150
236
Table 21
(continued)
BINOMIAL COEFFICIENTS
n! _ n(w - 1) - ■ • (n - A: + 1)
kl{n-k)\ " ft!
n
n — k
0! = 1
X
10
11
12
13
14
15
10
1
-
11
11
1
12
66
12
1
13
286
78
13
1
14
1001
S64
91
14
1
15
3003
1S6S
455
106
16
1
16
8008
4368
1820
660
120
16
17
19448
12376
6188
2380
680
136
18
43758
31824
18564
8568
3060
816
19
92378
75582
60388
27132
11628
3876
20
184756
167960
125970
77520
38760
15504
21
352716
352716
293930
203490
116280
54264
22
646646
705432
646646
497420
319770
170544
23
1144066
1362078
1352078
1144066
817190
490314
24
1961256
2496144
2704156
2496144
1961256
1307504
25
3268760
4457400
5200300
5200300
4457400
3268760
26
5311735
7726160
9657700
10400600
9657700
7726160
27
8436285
13037895
17383860
20058300
20058300
17383860
28
13123110
21474180
30421755
37442160
40116600
37442160
29
20030010
34597290
51895935
67863915
77558760
77558760
30
30045015
54627300
86493225
119759850
145422675
155117520
For k > 16 use the fact that
71
n — k
237
n
«2
n^
y/n
VlOn
y/n
Viow
'(
1/n
VlOOn
1
1
1
1.000 000
3.162 278
1.000 000
2.154 435
4.641 589
1.000 000
2
4
8
1.414 214
4.472 136
1.259 921
2.714 418
5.848 035
.500 000
3
9
27
1.732 051
5.477 226
1.442 250
3.107 233
6.694 330
.333 333
4
16
64
2.000 000
6.324 555
1.687 401
3.419 952
7.368 063
.250 000
5
25
125
2.236 068
7.071 068
1.709 976
3.684 031
7.937 005
.200 000
6
36
216
2.449 490
7.745 967
1.817 121
3.914 868
8.434 327
.166 667
7
49
343
2.645 751
8.366 600
1.912 931
4.121 285
8.879 040
.142 857
8
64
512
2.828 427
8.944 272
2.000 000
4.308 869
9.283 178
.125 000
9
81
729
3.000 000
9.486 833
2.080 084
4.481 405
9.654 894
.111 HI
:o
100
1 000
3.162 278
10.000 00
2.154 435
4.641 589
10.000 00
.100 000
11
121
1 331
3.316 625
10.488 09
2.223 980
4.791 420
10.322 80
.090 909
12
144
1 728
3.464 102
10.954 45
2.289 428
4.932 424
10.626 59
.083 333
13
169
2 197
3.005 551
11.401 75
2.351 335
5.065 797
10.913 93
.076 923
14
196
2 744
3.741 657
11.832 16
2.410 142
5.192 494
11.186 89
.071 429
15
225
3 375
3.872 983
12.247 45
2.466 212
5.313 293
11.447 14
.066 667
16
256
4 096
4.000 000
12.649 11
2.519 842
5.428 835
11.696 07
.062 500
17
289
4 913
4.123 106
13.038 40
2.571 282
5.539 658
11.934 83
.058 824
18
324
5 832
4.242 641
13.416 41
2.620 741
5.646 216
12.164 40
.055 556
19
361
6 859
4.358 899
13.784 05
2.668 402
5.748 897
12.385 62
.052 632
20
400
8 000
4.472 136
14.142 14
2.714 418
5.848 035
12.599 21
.050 000
21
441
9 261
4.582 576
14.491 38
2.758 924
5.943 922
12.805 79
.047 619
22
484
10 648
4.690 416
14.832 40
2.802 039
6.036 811
13.005 91
.045 455
23
529
12 167
4.795 832
15.165 75
2.843 867
6.126 926
13.200 06
.043 478
24
576
13 824
4.898 979
15.491 93
2.884 499
6.214 465
13.388 66
.041 667
25
625
15 625
5.000 000
15.811 39
2.924 018
6.299 605
13.572 09
.040 000
26
676
17 576
5.099 020
16.124 52
2.962 496
6.382 504
13.750 69
.038 462
27
729
19 683
5.196 152
16.431 68
3.000 000
6.463 304
13.924 77
.037 037
28
784
21952
5.291 503
16.733 20
3.036 589
6.542 133
14.094 60
.035 714
29
841
24 389
5.385 165
17.029 39
3.072 317
6.619 106
14.260 43
.034 483
SO
900
27 000
5.477 226
17.320 51
3.107 233
6.694 330
14.422 50
.033 333
31
961
29 791
5.567 764
17.606 82
3.141 381
6.767 899
14.581 00
.032 258
32
1024
32 768
5.656 854
17.888 54
3.174 802
6.839 904
14.736 13
.031 250
33
1 089
35 937
5.744 563
18.165 90
3.207 534
6.910 423
14.888 06
.030 303
34
1 156
39 304
5.830 952
18.439 09
3.239 612
6.979 532
15.036 95
.029 412
35
1225
42 875
5.916 080
18.708 29
3.271 066
7.047 299
15.182 94
.028 571
36
1 296
46 656
6.000 000
18.973 67
3.301 927
7.113 787
15.326 19
.027 778
37
1 369
50 653
6.082 763
19.235 38
3.332 222
7.179 054
15.466 80
.027 027
38
1 444
54 872
6.164 414
19.493 59
3.361 975
7.243 156
15.604 91
.026 316
39
1 521
59 319
6.244 998
19.748 42
3.391 211
7.306 144
15.740 61
.025 641
40
1 600
64 000
6.324 555
20.000 00
3.419 952
7.368 063
15.874 01
.025 000
41
1 681
68 921
6.403 124
20.248 46
3.448 217
7.428 959
16.005 21
.024 390
42
1 764
74 088
6.480 741
20.493 90
3.476 027
7.488 872
16.134 29
.023 810
43
1 849
79 507
6.557 439
20.736 44
3.503 398
7.547 842
16.261 33
.023 256
44
1 936
85 184
6.633 250
20.976 18
3.530 348
7.605 905
16.386 43
.022 727
45
2 025
91 125
6.708 204
21.213 20
3.556 893
7.663 094
16.509 64
.022 222
46
2 116
97 336
6.782 330
21.447 61
3.583 048
7.719 443
16.631 03
.021 739
47
2 209
103 823
6.855 655
21.679 48
3.608 826
7.774 980
16.750 69
.021 277
48
2 304
110 592
6.928 203
21.908 90
3.634 241
7.829 735
16.868 65
.020 833
49
2 401
117 649
7.000 000
22.135 94
3.659 306
7.883 735
16.984 99
.020 408
50
2 500
125 000
7.071 068
22.360 68
3.684 031
7.937 005
17.099 76
.020 000
238
Tabre 22
(continued)
SQUARES, CUBES, ROOTS AND RECIPROCALS
n
n2
„3
\n
VlOn
■\/n
V^lOw
\/lOOn
1/n
50
2 500
125 000
7.071 068
22.360 68
3.684 031
7.937 005
17.099 76
.020 000
51
2 601
132 651
7.141 428
22.583 18
3.708 430
7.989 570
17.213 01
.019 608
52
2 704
140 608
7.211 103
22.803 51
3.732 511
8.041 452
17.324 78
.019 231
53
2 809
148 877
7.280 110
23.021 73
3.756 286
8.092 672
17.435 13
.018 868
54
2 916
157 464
7.348 469
23.237 90
3.779 763
8.143 253
17.544 11
.018 519
55
3 025
166 375
7.416 198
23.452 08
3.802 952
8.193 213
17.651 74
.018 182
56
3 136
175 616
7.483 315
23.664 32
3.825 862
8.242 571
17.758 08
.017 857
57
3 249
185 193
7.549 834
23.874 67
3.848 501
8.291 344
17.863 16
.017 544
58
3 364
195 112
7.615 773
24.083 19
3.870 877
8.339 551
17.967 02
.017 241
59
3 481
205 379
7.681 146
24.289 92
3.892 996
8.387 207
18.069 69
.016 949
60
3 600
216 000
7.745 967
24.494 90
3.914 868
8.434 327
18.171 21
.016 667
61
3 721
226 981
7.810 250
24.698 18
3.936 497
8.480 926
18.271 60
.016 393
62
3 844
238 328
7.874 008
24.899 80
3.957 892
8.527 019
18.370 91
.016 129
63
3 969
250 047
7.937 254
25.099 80
3.979 057
8.572 619
18.469 15
,015 873
64
4 096
262 144
8.000 000
25.298 22
4.000 000
8.617 739
18.566 36
.015 625
65
4 225
274 625
8.062 258
25.495 10
4.020 726
8.662 391
18.662 56
.015 385
66
4 356
287 496
8.124 038
25.690 47
4.041 240
8.706 588
18.757 77
.015 152
67
4 489
300 763
8.185 353
25.884 36
4.061 548
8.750 340
18.852 04
.014 925
68
4 624
314 432
8.246 211
26.076 81
4.081 655
8.793 659
18.945 36
.014 706
69
4 761
328 509
8.306 624
26.267 85
4.101 566
8.836 556
19.037 78
.014 493
70
4 900
343 000
8.366 600
26.457 51
4.121 285
8.879 040
19.129 31
.014 286
71
5 041
357 911
8.426 150
26.645 83
4.140 818
8.921 121
19.219 97
.014 085
72
5 184
373 248
8.485 281
26.832 82
4.160 168
8.962 809
19.309 79
.013 889
73
5 329
389 017
8.544 004
27.018 51
4.179 339
9.004 113
19.398 77
.013 699
74
5 476
405 224
8.602 325
27.202 94
4.198 336
9.045 042
19.486 95
.013 514
75
5 625
421 875
8.660 254
27.386 13
4.217 163
9.085 603
19.574 34
.013 333
76
5 776
438 976
8.717 798
27.568 10
4.235 824
9.125 805
19.660 95
.013 158
77
5 929
456 533
8.774 964
27.748 87
4.254 321
9.165 656
19.746 81
.012 987
78
6 084
474 552
8.831 761
27.928 48
4.272 659
9.205 164
19.831 92
.012 821
79
6 241
493 039
8.888 194
28.106 94
4.290 840
9.244 335
19.916 32
.012 6.58
80
6 400
512 000
8.944 272
28.284 27
4.308 869
9.283 178
20.000 00
.012 500
81
6 561
531 441
9.000 000
28.460 50
4.326 749
9.321 698
20.082 99
.012 346
82
6 724
551 368
9.055 385
28.635 64
4.344 481
9.359 902
20.165 30
.012 195
83
6 889
571 787
9.110 434
28.809 72
4.362 071
9.397 796
20.246 94
.012 048
84
7 056
592 704
9.165 151
28.982 75
4.379 519
9.435 388
20.327 93
.011 905
85
7 225
614 125
9.219 544
29.154 76
4.396 830
9.472 682
20.408 28
.011 765
86
7 396
636 056
9.273 618
29.325 76
4.414 005
9.509 685
20.488 00
.011 628
87
7 569
658 503
9.327 379
29.495 76
4.431 048
9.546 403
20.567 10
.011 494
88
7 744
681 472
9.380 832
29.664 79
4.447 960
9.582 840
20.645 60
.011 364
89
7 921
704 969
9.433 981
29.832 87
4.464 745
9.619 002
20.723 51
.011 236
90
8 100
729 000
9.486 833
30.000 00
4.481 405
9.654 894
20.800 84
.011 111
91
8 281
753 571
9.539 392
30.166 21
4.497 941
9.690 521
20.877 59
.010 989
92
8 464
778 688
9.591 663
30.331 50
4.514 357
9.725 888
20.953 79
.010 870
93
8 649
804 357
9.643 651
30.495 90
4.530 655
9.761 000
21.029 44
.010 753
94
8 836
830 584
9.695 360
30.659 42
4.546 836
9.795 861
21.104 54
.010 638
95
9 025
857 375
9.746 794
30.822 07
4.562 903
9.830 476
21.179 12
.010 526
96
9 216
884 736
9.797 959
30.983 87
4.578 857
9.864 848
21.253 17
.010 417
97
9 409
912 673
9.848 858
31.144 82
4.594 701
9.898 983
21.326 71
.010 309
98
9 604
941 192
9.899 495
31.304 95
4.610 436
9.932 884
21.399 75
.010 204
99
9 801
970 299
9.949 874
31.464 27
4.626 065
9.966 555
21.472 29
.010 101
100
10 000
1 000 000
10.00 000
31.622 78
4.641 589
10.00 000
21.544 35
.010 000
239
COMPOUND AMOUNT: (1 + r)"
If a principal P is deposited at interest rate r (in decimals) compounded
annually, then at the end of n years the accumulated amount A = Pd+r)-*.
\
1^.
11%
1|%
2%
2^%
3%
A^'
5%
6%
1
1.0100
1.0125
1.0150
1.0200
1.0250
1.0300
1.0400
1.0500
1.0600
2
1.0201
1.0252
1.0302
1.0404
1.0506
1.0609
1.0816
1.1025
1.1236
3
1.0303
1.0380
1.0457
1.0612
1.0769
1.0927
1.1249
1.1576
1.1910
4
1.0406
1.0509
1.0614
1.0824
1.1038
1.1255
1.1699
1.2155
1.2635
5
1.0510
1.0641
1.0773
1.1041
1.1314
1.1593
1.2167
1.2763
1.3382
6
1.0615
1.0774
1.0934
1.1262
1.1597
1.1941
1.2653
1.3401
1.4185
7
1.0721
1.0909
1.1098
1.1487
1.1887
1.2299
1.3159
1.4071
1.5036
8
1.0829
1.1045
1.1265
1.1717
1.2184
1.2668
1.3688
1.4775
1.5938
9
1.0937
1.1183
1.1434
1.1951
1.2489
1.3048
1.4233
1.5513
1.6895
10
1.1046
1.1323
1.1605
1.2190
1.2801
1.3439
1.4802
1.6289
1.7908
11
1.1157
1.1464
1.1779
1.2434
1.3121
1.3842
1.53iJ5
1.7103
1.8983
12
1.1268
1.1608
1.1956
1.2682
1.3449
1.4258
1.6010
1.7959
2.0122
13
1.1381
1.1753
1.2136
1.2936
1.3785
1.4685
1.6651
1.8856
2.1329
14
1.1495
1.1900
1.2318
1.3195
1.4130
1.5126
1.7317
1.9799
2.2609
15
1.1610
1.2048
1.2502
1.3459
1.4483
1.5580
1.8009
2.0789
2.3966
16
1.1726
1.2199
1.2690
1.3728
1.4845
1.6047
1.8730
2.1829
2.5404
17
1.1843
1.2351
1.2880
1.4002
1.5216
1.6528
1.9479
2.2920
2.6928
18
1.1961
1.2506
1.3073
1.4282
1.5597
1.7024
2.0258
2.4066
2.8543
19
1.2081
1.2662
1.3270
1.4568
1.5987
1.7535
2.1068
2.5270
3.0256
20
1.2202
1.2820
1.3469
1.4859
1.6386
1.8061
2.1911
2.6533
3.2071
21
1.2324
1.2981
1.3671
1.5157
1.6796
1.8603
2.2788
2.7860
3.3996
22
1.2447
1.3143
1.3876
1.5460
1.7216
1.9161
2.3699
2.9253
3.6035
23
1.2572
1.3307
1.4084
1.5769
1.7646
1.9736
2.4647
3.0715
3.8197
24
1.2697
1.3474
1.4295
1.6084
1.8087
2.0328
2.5633
3.2251
4.0489
25
1.2824
1.3642
1.4509
1.6406
1.8539
2.0938
2.6658
3.3864
4.2919
26
1.2953
1.3812
1.4727
1.6734
1.9003
2.1566
2.7725
3.5557
4.5494
27
1.3082
1.3985
1.4948
1.7069
1.9478
2.2213
2.8834
3.7335
4.8223
28
1.3213
1.4160
1.5172
1.7410
1.9965
2.2879
2.9987
3.9201
5.1117
29
1.3345
1.4337
1.5400
1.7758
2.0464
2.3566
3.1187
4.1161
5.4184
30
1.3478
1.4516
1.6631
1.8114
2.0976
2.4273
3.2434
4.3219
5.7435
31
1.3613
1.4698
1.5865
1.8476
2,1500
2.5001
3.3731
4.5380
6.0881
32
1.3749
1.4881
1.6103
1.8845
2,2038
2.5751
3.5081
4.7649
6.4534
33
1.3887
1.5067
1.6345
1.9222
2,2589
2.6523
3.6484
5.0032
6.8406
34
1.4026
1.5256
1.6590
1.9607
2.3153
2.7319
3.7943
6.2533
7.2510
35
1.4166
1.5446
1.6839
1.9999
2.3732
2.8139
3.9461
5.5160
7.6861
36
1.4308
1.5639
1.7091
2.0399
2.4325
2.8983
4.1039
5.7918
8.1473
37
1.4451
1.5835
1.7348
2.0807
2.4933
2.9852
4.2681
6.0814
8.6361
38
1.4595
1.6033
1.7608
2.1223
2,6557
3.0748
4.4388
6.3855
9.1543
39
1.4741
1.6233
1.7872
2.1647
2.6196
3.1670
4.6164
6.7048
9.7035
40
1.4889
1.6436
1.8140
2.2080
2.6851
3.2620
4.8010
7.0400
10.2857
41
1.5038
1.6642
1.8412
2.2522
2.7522
3.3599
4.9931
7.3920
10,9029
42
1.5188
1.6850
1.8688
2.2972
2.8210
3.4607
5.1928
7.7616
11.5570
43
1.5340
1.7060
1,8969
2.3432
2.8915
3.5645
5.4005
8.1497
12.2505
44
1.5493
1.7274
1.9253
2.3901
2.9638
3.6715
5.6165
8.5572
12.9855
45
1.5648
1.7489
1.9542
2.4379
3.0379
3.7816
5.8412
8.9850
13.7646
46
1.5805
1.7708
1.9835
2.4866
3.1139
3.8950
6.0748
9.4343
14,5905
47
1.5963
1.7929
2.0133
2.5363
3.1917
4.0119
6.3178
9.9060
15.4659
48
1.6122
1.8154
2.0435
2.5871
3.2715
4.1323
6.5705
10.4013
16.3939
49
1.6283
1.8380
2.0741
2.6388
3.3533
4.2562
6.8333
10.9213
17.3775
50
1.6446
1.8610
2.1052
2.6916
3.4371
4.3839
7.1067
11.4674
18.4202
240
TABLE
24
PRESENT VALUE OF AN AMOUNT: (1 + r) "
The present value P which will amount to A in « years at an in-
terest rate of r (in decimals) compounded annually is P =^ -A(l + r)".
1%
li%
i^-%
2%
H'
3%
4'c
6%
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
.99010
.98030
.97059
.96098
.95147
.94205
.93272
.92348
.91434
.90529
.89632
.88745
.87866
.86996
.86135
.85282
.84438
.83602
.82774
.81954
.81143
.80340
.79544
.78757
.77977
.77205
.76440
.75684
.74934
.74192
.73458
.72730
.72010
.71297
.70591
.69892
.69200
.68515
.67837
.67165
.66500
.65842
.65190
.64545
.63905
.63273
.62646
.62026
.61412
.60804
.98765
.97546
.96342
.95152
.93978
.92817
.91672
.90540
.89422
.88318
.87228
.86151
.85087
.84037
.82999
.81975
.80963
.79963
.78976
.78001
.77038
.76087
.75147
.74220
.73303
.72398
.71505
.70622
.69750
.68889
.68038
.67198
.66369
.65549
.64740
.63941
.63152
.62372
.61602
.60841
.60090
.59348
.58616
.57892
.57177
.56471
.55774
.55086
.54406
.53734
.98522
.97066
.95632
.94218
.92826
.91454
.90103
.88771
.87459
.86167
.84893
.83639
.82403
.81185
.79985
.78803
.77639
.76491
.75361
.74247
.73150
.72069
.71004
.69954
.68921
.67902
.66899
.65910
.64936
.63976
.63031
.62099
.61182
.60277
.59387
.58509
.57644
.56792
.55953
.55126
.54312
.53509
.52718
.51939
.51171
.50415
.49670
.48936
.48213
.47500
.98039
.96117
.94232
.92385
.90573
.88797
.87056
.85349
.83676
.82035
.80426
.78849
.77303
.75788
.74301
.72845
.71416
.70016
.68643
.67297
.65978
.64684
.63416
.62172
.60953
.59758
.58586
.57437
.56311
.55207
.54125
.53063
.52023
.51003
.50003
.49022
.48061
.47119
.46195
.45289
.44401
.43530
.42677
.41840
.41020
.40215
.39427
.38654
.37896
.37153
.97561
.95181
.92860
.90595
.88385
.86230
.84127
.82075
.80073
.78120
.76214
.74356
.72542
.70773
.69047
.67362
.65720
.64117
.62553
.61027
.59539
.58086
.56670
.55288
.53939
.52623
.51340
.50088
.48866
.47674
.46511
.45377
.44270
.43191
.42137
.41109
.40107
.39128
.38174
.37243
.36335
.35448
.34584
.33740
.32917
.32115
.31331
.30567
.29822
.29094
.97087
.94260
.91514
.88849
.86261
.83748
.81309
.78941
.76642
.74409
.72242
.70138
.68095
.66112
.64186
.62317
.60502
.58739
.57029
.55368
.53755
.52189
.50669
.49193
.47761
.46369
.45019
.43708
.42435
.41199
.39999
.38834
.37703
.36604
.35538
.34503
.33498
.32523
.31575
.30656
.29763
.28896
.28054
.27237
.26444
.25674
.24926
.24200
.23495
.22811
.96154
.92456
.88900
.85480
.82193
.79031
.75992
.73069
.70259
.67556
.64958
.62460
.60057
.57748
.55526
.53391
.51337
.49363
.47464
.45639
.43883
.42196
.40573
.39012
.37512
.36069
.34682
.33348
.32065
.30832
.29646
.28506
.27409
.26355
.26342
.24367
.23430
.22529
.21662
.20829
.20028
.19257
.18517
.17805
.17120
.16461
.15828
.15219
.14634
.14071
.95238
.90703
.86384
.82270
.78353
.74622
.71068
.67684
.64461
.61391
.58468
.55684
.53032
.50507
.48102
.45811
.43630
.41552
.39573
.37689
.35894
.34185
.32557
.31007
.29530
.28124
.26785
.25509
.24295
.23138
.22036
.20987
.19987
.19035
.18129
.17266
.16444
.15661
.14915
.14205
.13528
.12884
.12270
.11686
.11130
.10600
.10095
.09614
.09156
.08720
.94340
.89000
.83962
.79209
.74726
.70496
.66506
.62741
.59190
.55839
.52679
.49697
.46884
.44230
.41727
.39365
.37136
.35034
.33051
.31180
.29416
.27751
.26180
.24698
.23300
.21981
.20737
.19563
.18456
.17411
.16425
.15496
.14619
.13791
.13011
.12274
.11579
.10924
.10306
.09722
.09172
.08653
.08163
.07701
.07265
.06854
.06466
.06100
.05755
.05429
241
AMOUNT OF AN ANNUITY:
(l + r)«-l
If a principal P is deposited at the end of each year at interest rate r (in
decimals) compounded annually, then at the end of n years the accumulated
amount is P . The process is often called an annuity.
\
1%
H%
u%
2%
2^%
3%
4fc
5%
6%
1
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
2
2.0100
2.0125
2.0150
2.0200
2-0250
2.0300
2.0400
2.0500
2.0600
3
3.0301
3.0377
3.0452
3.0604
3.0756
3.0909
3.1216
3.1525
3.1836
4
4.0604
4.0756
4.0909
4.1216
4.1525
4.1836
4.2465
4.3101
4.3746
5
5.1010
5.1266
5.1523
5.2040
5.2563
5.3091
5.4163
5.5256
5.6371
6
6.1520
6.1907
6.2296
6.3081
6.3877
6.4684
6.6330
6.8019
6.9753
7
7.2135
7.2680
7.3230
7.4343
7.5474
7.6625
7.8983
8.1420
8.3938
8
8.2857
8.3589
8.4328
8.5830
8.7361
8.8923
9.2142
9.5491
9.8975
9
9.3685
9.4634
9.5593
9.7546
9.9545
10.1591
10.5828
11.0266
11.4913
10
10.4622
10.5817
10.7027
10.9497
11.2034
11.4639
12.0061
12.5779
13.1808
11
11.5668
11.7139
11.8633
12.1687
12.4835
12.8078
13.4864
14.2068
14.9716
12
12.6825
12.8604
13.0412
13.4121
13.7956
14.1920
15.0258
15.9171
16.8699
13
13.8093
14.0211
14.2368
14.6803
15.1404
15.6178
16.62G8
17.7130
18.8821
14
14.9474
15.1964
15.4504
15.9739
16.5190
17.0863
18.2919
19.5986
21.0151
15
16.0969
16.3863
16.6821
17.2934
17.9319
18.5989
20.0236
21.5786
23.2760
16
17.2579
17.5912
17.9324
18.6393
19.3802
20.1569
21.8245
23.6575
25.6725
17
18.4304
18.8111
19.2014
20.0121
20.8647
21.7616
23.6975
25.8404
28.2129
18
19.6147
20.0462
20.4894
21.4123
22.3863
23.4144
25.6454
28.1324
30.9057
19
20.8109
21.2968
21.7967
22.8406
23.9460
25.1169
27.6712
30.5390
33.7600
20
22.0190
22.5630
23.1237
24.2974
25.5447
26.8704
29.7781
33.0660
36.7856
21
23.2392
23.8450
24.4705
25.7833
27.1833
28.6765
31.9692
35.7193
39.9927
22
24.4716
25.1431
25.8376
27.2990
28.8629
30.5368
34.2480
38.5052
43.3923
23
25.7163
26.4574
27.2251
28.8450
30.5844
32.4529
36.6179
41.4305
46.9958
24
26.9735
27.7881
28.6335
30.4219
32.3490
34.4265
39.0826
44.5020
50.8156
25
28.2432
29.1354
30.0630
32.0303
34.1578
36.4593
41.6459
47.7271
54.8645
26
29.5256
30.4996
31.5140
33.6709
36.0117
38.5530
44.3117
51.1135
59.1564
27
30.8209
31.8809
32.9867
35.3443
37.9120
40.7096
47.0842
54.6691
63.7058
28
32.1291
33.2794
34.4815
37.0512
39.8598
42.9309
49.9676
58.4026
68.5281
29
33.4504
34.6954
35.9987
38.7922
41.8563
45.2189
52.9663
62.3227
73.6398
30
34.7849
36.1291
37.5387
40.5681
43.9027
47.5754
66.0849
66.4388
79.0582
31
36.1327
37.5807
39.1018
42.3794
46.0003
50.0027
59.3283
70.7608
84.8017
32
37.4941
39.0504
40.6883
44.2270
48.1503
52.6028
62.7015
75.2988
90.8898
33
38.8690
40.5386
42.2986
46.1116
50.3540
55.0778
66.2095
80.0638
97.3432
34
40.2577
42.0453
43.9331
48.0338
52.6129
57.7302
69.8579
85.0670
104.1838
35
41.6603
43.5709
45.5921
49.9945
54.9282
60.4621
73.6522
90.3203
111.4348
36
43.0769
45.1155
47.2760
51.9944
57.3014
63.2759
77.5983
95.8363
119.1209
37
44.5076
46.6794
48.9851
54.0343
59.7339
66.1742
81.7022
101.6281
127.2681
38
45.9527
48.2629
50.7199
56.1149
62.2273
69.1594
85.9703
107.7095
135.9042
39
47.4123
49.8662
52.4807
58.2372
64.7830
72.2342
90.4091
114.0950
145.0585
40
48.8864
51.4896
54.2679
60.4020
67.4026
75.4013
95.0255
120.7998
154.7620
41
50.3752
53.1332
56.0819
62.6100
70.0876
78.6633
99.8265
127.8398
165.0477
42
51.8790
54.7973
57.9231
64.8622
72.8398
82.0232
104.8196
135.2318
175.9505
43
53.3978
56.4823
59.7920
67.1595
75.6608
85.4839
110.0124
142.9933
187.5076
44
54.9318
58.1883
61.6889
69.5027
78.5523
89.0484
115.4129
151.1430
199.7580
4S
56.4811
59.9157
63.6142
71.8927
81.5161
92.7199
121.0294
159.7002
212.7435
46
58.0459
61.6646
65.5684
74.3306
84.5540
96.5015
126.8706
168.6852
226.5081
47
59.6263
63.4354
67.5519
76.8172
87.6679
100.3965
132.9454
178.1194
241.0986
48
61.2226
65.2284
69.5652
79.3535
90.8596
104.4084
139.2632
188.0254
256.5645
49
62.8348
67.0437
71.6087
81.9406
94.1311
108.5406
145.8337
198.4267
272.9584
50
64.4632
68.8818
73.6828
84.5794
97.4843
112.7969
152.6671
209.3480
290.3359
242
TABLE
26
PRESENT VALUE OF AN ANNUITY:
1- (l + r)-"
An annuity in which the yearly payment at the end of each of n years is A at an
Tl — (1 -f- r)~"
interest rate r (in decimals) compounded annually has present value A
1%
U^-.
1|%
2%
21%
3%
4%
5%
6%
1
0.9901
0.9877
0.9852
0.9804
0.9756
0.9709
0.9615
0.9524
0.9434
2
1.9704
1.9631
1.9559
1.9416
1.9274
1.9135
1.8861
1.8594
1.8334
3
2.9410
2.9265
2.9122
2.8839
2.8560
2.8286
2.7751
2.7232
2.6730
4
3.9020
3.8781
3.8544
3.8077
3.7620
3.7171
3.6299
3.5460
3.4651
B
4.8534
4.8178
4.7826
4.7135
4.6458
4.5797
4.4518
4.3295
4.2124
6
5.7955
5.7460
5.6972
5.6014
5.5081
5.4172
5.2421
5.0757
4.9173
7
6.7282
6.6627
6.5982
6.4720
6.3494
6.2303
6.0021
5.7864
5.5824
8
7.6517
7.5681
7.4859
7.3255
7.1701
7.0197
6.7327
6.4632
6.2098
9
8.5660
8.4623
8.3605
8.1622
7.9709
7.7861
7.4353
7.1078
6.8017
10
9.4713
9.3455
9.2222
8.9826
8.7521
8.5302
8.1109
7.7217
7.3601
11
10.3676
10.2178
10.0711
9.7868
9.5142
9.2526
8.7605
8.3064
7.8869
12
11.2551
11.0793
10.9075
10.5753
10,2578
9.9540
9.3851
8.8633
8.3838
13
12.1337
11.9302
11.7315
11.3484
10.9832
10.6350
9.9856
9.3936
8.8527
14
13.0037
12.7706
12.5434
12.1062
11.6909
11.2961
10.5631
9.8986
9.2950
15
13.8651
13.6005
13.3432
12.8493
12.3814
11.9379
11.1184
10.3797
9.7122
16
14.7179
14.4203
14.1313
13.5777
13.0550
12.5611
11.6523
10.8378
10.1059
17
15.5623
15.2299
14.9076
14.2919
13.7122
13.1661
12.1657
11.2741
10.4773
18
16.3983
16.0295
15.6726
14.9920
14.3534
13.7535
12.6593
11.6896
10.8276
19
17.2260
16.8193
16.4262
15.6785
14.9789
14.3238
13.1339
12.0853
11.1581
20
18.0456
17.5993
17.1686
16.3514
15.5892
14.8775
13.5903
12.4622
11.4699
21
18.8570
18.3697
17.9001
17.0112
16.1845
15.4150
14.0292
12.8212
11.7641
22
19.6604
19.1306
18.6208
17.6580
16.7654
15.9369
14.4511
13.1630
12.0416
23
20.4558
19.8820
19.3309
18.2922
17.3321
16.4436
14.8568
13.4886
12.3034
24
21.2434
20.6242
20.0304
18.9139
17.8850
16.9355
15.2470
13.7986
12.5504
25
22.0232
21.3573
20.7196
19.5235
18.4244
17.4131
15.6221
14.0939
12.7834
26
22.7952
22.0813
21.3986
20.1210
18.9506
17.8768
15.9828
14.3752
13.0032
27
23.5596
22.7963
22.0676
20.7069
19.4640
18.3270
16.3296
14.6430
13.2105
28
24.3164
23.5025
22.7267
21.2813
19.9649
18.7641
16.6631
14.8981
13.4062
29
25.0658
24.2000
23.3761
21.8444
20.4535
19.1885
16.9837
15.1411
13.5907
30
25.8077
24.8889
24.0158
22.3965
20.9303
19.6004
17.2920
15.3725
13.7648
31
26.5423
25.5693
24.6461
22.9377
21.3954
20.0004
17.5885
15.5928
13.9291
32
27.2696
26.2413
25.2671
23.4683
21.8492
20.3888
17.8736
15.8027
14.0840
33
27.9897
26.9050
25.8790
23.9886
22.2919
20.7658
18.1476
16.0025
14.2302
34
28.7027
27.5605
26.4817
24.4986
22.7238
21.1318
18.4112
16.1929
14.3681
35
29.4086
28.2079
27.0756
24.9986
23.1452
21.4872
18.6646
16.3742
14.4982
36
30.1075
28.8473
27.6607
25.4888
23.5563
21.8323
18.9083
16.5469
14.6210
37
30.7995
29.4788
28.2371
25.9695
23.9573
22.1672
19.1426
16.7113
14.7368
38
31.4847
30.1025
28.8051
26.4406
24.3486
22.4925
19.3679
16.8679
14.8460
39
32.1630
30.7185
29.3646
26.9026
24.7303
22.8082
19.5845
17.0170
14.9491
40
32.8347
31.3269
29.9158
27.3555
25.1028
23.1148
19.7928
17.1591
15.0463
41
33.4997
31.9278
30.4590
27.7995
25.4661
23.4124
19.9931
17.2944
15.1380
42
34.1581
32.5213
30.9941
28.2348
25.8206
23.7014
20.1856
17.4232
15.2245
43
34.8100
33.1075
31.5212
28.6616
26.1664
23.9819
20.3708
17.5459
15.3062
44
35.4555
33.6864
32.0406
29.0800
26.5038
24.2543
20.5488
17.6628
15.3832
45
36.0945
34.2582
32.5523
29.4902
26.8330
24.5187
20.7200
17.7741
15.4558
46
36.7272
34.8229
33.0565
29.8923
27.1542
24.7754
20.8847
17.8801
15.5244
47
37.3537
35.3806
33.5532
30.2866
27.4675
26.0247
21.0429
17.9810
15.5890
48
37.9740
35.9315
34.0426
30.6731
27.7732
25.2667
21.1951
18.0772
15.6500
49
38.5881
36.4755
34.5247
31.0521
28.0714
25.5017
21.3415
18.1687
15.7076
50
39.1961
37.0129
34.9997
31.4236
28.3623
25.7298
21.4822
18.2559
15.7619
243
X
1
2
3
4
5
6
7
8
9
0.
1.0000
.9975
.9900
.9776
.9604
.9385
.9120
.8812
.8463
.8075
1.
.7652
.7196
.6711
.6201
.5669
.5118
.4554
.3980
.3400
.2818
2.
.2239
.1666
.1104
.0555
.0025
-.0484
-.0968
-.1424
-.1850
-.2243
3.
-.2601
-.2921
-.3202
-.3443
-.3643
-.3801
-.3918
-.3992
-.4026
-.4018
4.
-.3971
-.3887
-.3766
-.3610
-.3423
-.3205
-.2961
-.2693
-.2404
-.2097
5.
-.1776
-.1443
-.1103
-.0758
-.0412
-.0068
.0270
.0599
.0917
.1220
6.
.1506
.1773
.2017
.2238
.2433
.2601
.2740
.2851
.2931
.2981
7.
.3001
.2991
.2951
.2882
.2786
.2663
.2516
.2346
.2154
.1944
8.
.1717
.1475
.1222
.0960
.0692
.0419
.0146
-.0125
-.0392
-.0653
9.
-.0903
-.1142
-.1367
-.1577
-.1768
-.1939
-.2090
-.2218
-.2323
-.2403
X
1
2
3
4
5
6
7
8
9
0.
.0000
.0499
.0995
.1483
.1960
.2423
.2867
.3290
.3688
.4059
1.
.4401
.4709
.4983
.5220
.5419
.5579
.5699
.5778
.5815
.5812
2.
.5767
.6683
.5660
.5399
.5202
.4971
.4708
.4416
.4097
.3754
3.
.3391
.3009
.2613
.2207
.1792
.1374
.0955
.0538
.0128
-.0272
4.
-.0660
-.1033
-.1386
-.1719
-.2028
-.2311
-.2566
-.2791
-.2985
-.3147
6.
-.3276
-.3371
-.3432
-.3460
-.3453
-.3414
-.3343
-.3241
-.3110
-.2951
6.
-.2767
-.2559
-.2329
-.2081
-.1816
-.1538
-.1250
-.0953
-.0652
-.0349
7.
-.0047
.0252
.0543
.0826
.1096
.1352
.1592
.1813
.2014
.2192
8.
.2346
.2476
.2580
.2657
.2708
.2731
.2728
.2697
.2641
.2559
9.
.2453
.2324
.2174
.2004
.1816
.1613
.1395
.1166
.0928
.0684
244
X
1
2
3
4
5
6
7
8
9
0.
00
-1.5342
-1.0811
-.8073
-.6060
-.4445
-.3085
-.1907
-.0868
.0056
1.
.0883
.1622
.2281
.2865
.3379
.3824
.4204
.4520
.4774
.4968
2.
.5104
.5183
.5208
.5181
.5104
.4981
.4813
.4605
.4359
.4079
3.
.3769
.3431
.3071
.2691
.2296
.1890
.1477
.1061
.0645
.0234
4.
-.0169
-.0561
-.0938
-.1296
-.1633
-.1947
-.2235
-.2494
-.2723
-.2921
6.
-.3085
-.3216
-.3313
-.3374
-.3402
-.3395
-.3354
-.3282
-.3177
-.3044
6.
-.2882
-.2694
-.2483
-.2251
-.1999
-.1732
-.1452
-.1162
-.0864
-.0563
7.
-.0259
.0042
.0339
.0628
.0907
.1173
.1424
.1658
.1872
.2065
8.
.2235
.2381
.2501
.2595
.2662
.2702
.2715
.2700
.2659
.2592
9.
.2499
.2383
.2245
.2086
.1907
.1712
.1502
.1279
.1045
.0804
X
1
2
3
4
5
6
7
8
9
0.
— eo
-6.4590
-3.3238
-2.2931
-1.7809
-1.4715
-1.2604
-1.1032
-.9781
-.8731
1.
-.7812
-.6981
-.6211
-.5485
-.4791
-.4123
-.3476
-.2847
-.2237
-.1644
2.
-.1070
-.0517
.0015
.0523
.1005
.1459
.1884
.2276
.2635
.2959
3.
.3247
.3496
.3707
.3879
.4010
.4102
.4154
.4167
.4141
,4078
4.
.3979
.3846
.3680
.3484
.3260
.3010
.2737
.2445
.2136
.1812
5.
.1479
.1137
.0792
.0445
.0101
-.0238
-.0568
-.0887
-.1192
-.1481
6.
-.1750
-.1998
-.2223
-.2422
-.2596
-.2741
-.2857
-.2945
-.3002
-.3029
7.
-.3027
-.2995
-.2934
-.2846
-.2731
-.2591
-.2428
-.2243
-.2039
-.1817
8.
-.1581
-.1331
-.1072
-.0806
-.0535
-.0262
.0011
.0280
.0544
.0799
9.
.1043
.1275
.1491
.1691
.1871
.2032
.2171
.2287
.2379
.2447
245
X
1
2
3
4
5
6
7
8
9
0.
1.000
1.003
1.010
1.023
1.040
1.063
1.092
1.126
1.167
1.213
1.
1.266
1.326
1.394
1.469
1.553
1.647
1.750
1.864
1.990
2.128
2.
2.280
2.446
2.629
2.830
3.049
3.290
3.553
3.842
4.157
4.503
3.
4.881
5.294
5.747
6.243
6.785
7.378
8.028
8.739
9.517
10.37
4.
11.30
12.32
13.44
14.67
16.01
17.48
19.09
20.86
22.79
24.91
5.
27.24
29.79
32.58
35.65
39.01
42.69
46.74
51.17
56.04
61.38
6.
67.23
73.66
80.72
88.46
96.96
106.3
116.5
127.8
140.1
153.7
7.
168.6
185.0
202.9
222.7
244.3
268.2
294.3
323.1
354.7
389.4
8.
427.6
469.5
515.6
566.3
621.9
683.2
750.5
824.4
905.8
995.2
9.
1094
1202
1321
1461
1595
1753
1927
2119
2329
2561
BESSEL FUNCTIONS
lAx)
X
1
2
3
4
5
6
7
8
9
0.
.0000
.0501
.1005
.1517
.2040
.2579
.3137
.3719
.4329
.4971
1.
.5652
.6375
.7147
.7973
.8861
.9817
1.085
1.196
1.317
1.448
2.
1.591
1.745
1.914
2.098
2.298
2.517
2.755
3.016
3.301
3.613
3.
3.953
4.326
4.734
5.181
5.670
6.206
6.793
7.436
8.140
8.913
4.
9.759
10.69
11.71
12.82
14.05
15.39
16.86
18.48
20.25
22.20
5.
24.34
26.68
29.25
32.08
35.18
38.59
42.33
46.44
50.95
56.90
6.
61.34
67.32
73.89
81.10
89.03
97.74
107.3
117.8
129.4
142.1
7.
156.0
171.4
188.3
206.8
227.2
249.6
274.2
301.3
331.1
363.9
8.
399.9
439.5
483.0
531.0
583.7
641.6
705.4
775.5
852.7
937.5
9.
1031
1134
1247
1371
1508
16.58
1824
2006
2207
2428
246
X
1
2
3
4
5
6
7
8
9
0.
00
2.4271
1.7527
1.3725
1.1145
.9244
.7775
.6605
.5653
.4867
1.
.4210
.3656
.3185
.2782
.2437
.2138
.1880
.1655
.1459
.1288
2.
.1139
.1008
.08927
.07914
.07022
.06235
.05540
.04926
.04382
.03901
3.
.03474
.03095
.02759
.02461
.02196
.01960
.01750
.01563
.01397
.01248
4.
.01116
.029980
.028927
.027988
.027149
.026400
.025730
.025132
.024597
.024119
6.
.023691
.023308
.022966
.022659
.022385
.022139
.021918
.021721
.021544
.021386
6.
.021244
.021117
.021003
.039001
.038083
.037259
.036520
.035857
.035262
.034728
7.
.034248
.033817
.033431
.033084
.032772
.032492
.032240
.032014
.031811
.031629
8.
.031465
.031317
.031185
.031066
.049588
.0*8626
.0-*7761
.0''6983
.016283
.0''5654
9.
.o-isoss
.0^579
.0^4121
.0*3710
.0^3339
.o-isooe
.0''2706
.0''2436
.0^2193
.0*1975
X
1
2
3
4
5
6
7
8
9
0.
CO
9.8538
4.7760
3.0560
2.1844
1.6564
1.3028
1.0503
.8618
.7165
1.
.6019
.5098
.4346
.3725
.3208
.2774
.2406
.2094
.1826
.1597
2.
.1399
.1227
.1079
.09498
.08372
.07389
.06528
.05774
.05111
.04529
3.
.04016
.03563
.03164
.02812
.02500
.02224
.01979
.01763
.01571
.01400
4.
.01248
.01114
.029938
.028872
.027923
.027078
.026325
.025654
.025055
.024521
5.
.024045
.023619
.023239
.022900
.022597
.022326
.022083
.021866
.021673
.021499
6.
.021344
.021205
.021081
.039691
.038693
.037799
.036998
.036280
.035636
.035059
7.
.034542
.034078
.033662
.033288
.032953
.032653
.032383
.032141
.031924
.031729
8.
.031554
.031396
.031255
.031128
.031014
.0-'9120
.0*8200
.0*7374
.0*6631
.0*5964
9.
.0*5364
.0*4825
.0*4340
.0*3904
.0*3512
.0*3160
.0*2843
.0*2559
.0*2302
.0*2072
247
X
1
2
3
4
5
6
7
8
9
0.
1.0000
1.0000
1.0000
.9999
.9995
.9990
.9980
.9962
.9936
.9898
1.
.9844
.9771
.9676
.9554
.9401
.9211
.8979
.8700
.8367
.7975
2.
.7517
.6987
.6377
.5680
.4890
.4000
.3001
.1887
.06511
-.07137
3.
-.2214
-.3855
-.5644
-.7584
-.9680
-1.1936
-1.4353
-1.6933
-1.9674
-2.2576
4.
-2.5634
-2.8843
-3.2195
-3.5679
-3.9283
-4.2991
-4.6784
-5.0639
-5.4531
-5.8429
5.
-6.2301
-6.6107
-6.9803
-7.3344
-7.6674
-7.9736
-8.2466
-8.4794
-8.6644
-8.7937
6.
-8.8583
-8.8491
-8.7561
-8.5688
-8.2762
-7.8669
-7.3287
-6.6492
-5.8155
-4.8146
7.
-3.6329
-2.2571
-.6737
1.1308
3.1695
5.4550
7.9994
10.814
13.909
17.293
8.
20.974
24.957
29.245
33.840
38.788
43.936
49.423
55.187
61.210
67.469
9.
73.936
80.576
87.350
94.208
101.10
107.95
114.70
121.26
127.54
133.43
X
1
2
3
4
5
6
7
8
9
0.
.0000
.022500
.01000
.02250
.04000
.06249
.08998
.1224
.1599
.2023
1.
.2496
.3017
.3587
.4204
.4867
.5576
.6327
.7120
.7963
.8821
2.
.9723
1.0654
1.1610
1.2585
1.3575
1.4572
1.5669
1.6567
1.7529
1.8472
3.
1.9376
2.0228
2.1016
2.1723
2.2334
2.2832
2.3199
2.3413
2.3454
2.3300
4.
2.2927
2.2309
2.1422
2.0236
1.8726
1.6860
1.4610
1.1946
.8837
.5251
5.
.1160
-.3467
-.8658
-1.4443
-2.0845
-2.7890
-3.5597
-4.3986
-5.3068
-6.2854
6.
-7.3347
-8.4545
-9.6437
-10.901
-12.223
-13.607
-15.047
-16.538
-18.074
-19.644
7.
-21.239
-22.848
-24.456
-26.049
-27.609
-29.116
-30.548
-31.882
-33.092
-34.147
8.
-35.017
-35.667
-36.061
-36.159
-35.920
-35.298
-34.246
-32.714
-30.661
-28.003
9.
-24.713
-20.724
-15.976
-10.412
-3.9693
3.4106
11.787
21.218
31.758
43.459
248
BESSEL FUNCTIONS
Ker (x)
X
1
2
3
4
5
6
7
8
9
0.
CO
2.4205
1.7331
1.3372
1.0626
.8559
.6931
.5614
.4529
.3625
1.
.2867
.2228
.1689
.1235
.08513
.05293
.02603
.023691
-.01470
-.02966
2.
-.04166
-.05111
-.05834
-.06367
-.06737
-.06969
-.07083
-.07097
-.07030
-.06894
3.
-.06703
-.06468
-.06198
-.05903
-.05590
-.05264
-.04932
-.04597
-.04265
-.03937
4.
-.03618
-.03308
-.03011
-.02726
-.02456
-.02200
-.01960
-.01734
-.01525
-.01330
5.
-.01151
-.029865
-.028359
-.026989
-.025749
-.024632
-.023632
-.022740
-.021952
-.021258
6.
-.036630
-.031295
.033191
.036991
.021017
.021278
.021488
.021653
.021777
.021866
7.
.021922
.021951
.021956
.021940
.021907
.021860
.021800
.021731
.021655
.021572
8.
.021486
.021397
.021306
.021216
.021126
.021037
.039511
.038675
.037871
.037102
9.
.036372
.035681
.035030
.034422
.033855
.033330
.032846
.032402
.031996
.031628
X
1
2
3
4
5
6
7
8
9
0.
-.7864
-.7769
-.7581
-.7331
-.7038
-.6716
-.6374
-.6022
-.5664
-.5305
1.
-.4960
-.4601
-.4262
-.3933
-.3617
-.3314
-.3026
-.2752
-.2494
-.2251
2.
-.2024
-.1812
-.1614
-.1431
-.1262
-.1107
-.09644
-.08342
-.07167
-.06083
3.
-.06112
-.04240
-.03458
-.02762
-.02145
-.01600
-.01123
-.027077
-.023487
-.034108
4.
.022198
.024386
.026194
.027661
.028826
.029721
.01038
.01083
.01110
.01121
5.
.01119
.01105
.01082
.01051
.01014
.029716
.029255
.028766
.028268
.027739
6.
.O272I6
.026696
.026183
.025681
.025194
.024724
.024274
.023846
.023440
.023058
7.
.022700
.022366
.022057
.021770
.021507
.021267
.021048
.038498
.036714
.036117
8.
.033696
.032440
.031339
.0*3809
-.0M449
-.031149
-.031742
-.032233
-.032632
-.032949
9.
-.033192
-.033368
-.033486
-.033552
-.033574
-.033567
-.033608
-.033430
-.033829
-.033210
249
TABLE
39
VALUES FOR APPROXIMATE
ZEROS OF BESSEL FUNCTIONS
The following table lists the first few positive roots of various equations. Note that for all cases
listed the successive large roots differ approximately hy v = 3.14159. . . .
n =
n = l
71 ^ 2
71 = 3
71 — 4
71 = 5
n - 6
2.4048
3.8317
5.1356
6.3802
7.5883
8.7715
9.9361
5.5201
7.0156
8.4172
9.7610
11.0647
12.3386
13.5893
J^(x) =
8.6537
11.7915
10.1735
13.3237
11.6198
14.7960
13.0152
16.2235
14.3725
17.6160
15.7002
18.9801
17.0038
20.3208
14.9309
16.4706
17.9598
19.4094
20.8269
22.2178
23.5861
18.0711
19.6159
21.1170
22.5827
24.0190
25.4303
26.8202
0.8936
2.1971
3.3842
4.5270
5.6452
6.7472
7.8377
3.9577
5.4297
6.7938
8.0976
9.3616
10.5972
11.8110
Y^(x) =
7.0861
10.2223
8.5960
11.7492
10.0235
13.2100
11.3965
14.6231
12.7301
15.9996
14.0338
17.3471
15.3136
18.6707
13.3611
14.8974
16.3790
17.8185
19.2244
20.6029
21.9583
16.5009
18.0434
19.5390
20.9973
22.4248
23.8265
25.2062
0.0000
1.8412
3.0542
4.2012
5.3176
6.4156
7.5013
3.8317
5.3314
6.7061
8.0152
9.2824
10.5199
11.7349
^n(«) =
7.0156
10.1735
8.5363
11.7060
9.9695
13.1704
11.3459
14.5859
12.6819
15.9641
13.9872
17.3128
15.2682
18.6374
13.3237
14.8636
16.3475
17.7888
19.1960
20.5755
21.9317
16.4706
18.0155
19.5129
20.9725
22.4010
23.8036
25.1839
2.1971
3.6830
5.0026
6.2536
7.4649
8.6496
9.8148
5.4297
6.9415
8.3507
9.6988
11.0052
12.2809
13.5328
l'n(*) =
8.5960
11.7492
10.1234
13.2858
11.5742
14.7609
12.9724
16.1905
14.3317
17.5844
15.6608
18.9497
16.9655
20.2913
14.8974
16.4401
17.9313
19.3824
20.8011
22.1928
23.5619
18.0434
19.5902
21.0929
22.5598
23.9970
25.4091
26.7995
250
TABLE
40
EXPONENTIAL, SINE AND COSINE INTEGRALS
Ei{x) = C ^du, St{x) = r ^^^du, Ci{x) =: f
cos«
du
X
Ei{x)
Si{x)
Ciix)
.0
CO
.0000
eo
.5
.5598
.4931
.1778
1.0
.2194
.9461
-.3374
1.5
.1000
1.3247
-.4704
2.0
.04890
1.6054
-.4230
2.5
.02491
1.7785
-.2859
3.0
.01305
1.8487
-.1196
3.5
.026970
1.8331
.0321
4.0
.023779
1.7582
.1410
4.5
.022073
1.6541
.1935
5.0
.021148
1.5499
.1900
5.5
.036409
1.4687
.1421
6.0
.033601
1.4247
.0681
6.5
.032034
1.4218
-.0111
7.0
.031155
1.4546
-.0767
7.5
.0''6583
1.5107
-.1156
8.0
.0''3767
1.5742
-.1224
8.5
.0''2162
1.6296
-.09943
9.0
.0*1245
1.6650
-.05535
9.5
.057185
1.6745
-.022678
10.0
.054157
1.6583
.04546
251
TABLE
41
LEGENDRE POIYNOMIALS Pn{x)
[Po{x) = l, Piix) = x]
X
P2{^)
P3N
P^{^)
Pii'^)
.00
-.5000
.0000
.3750
.0000
.05
-.4963
-.0747
.3657
.0927
.10
-.4850
-.1475
.3379
.1788
.16
-.4663
-.2166
.2928
.2523
.20
-.4400
-.2800
.2320
.3075
.25
-.4063
-.3359
.1577
.3397
.80
-.3650
-.3825
.0729
.3454
.35
-.3163
-.4178
-.0187
.3225
.40
-.2600
-.4400
-.1130
.2706
.46
-.1963
-.4472
-.2050
.1917
.50
-.1250
-.4375
-.2891
.0898
.55
-.0463
-.4091
-.3690
-.0282
.60
.0400
-.3600
-.4080
-.1526
.65
.1338
-.2884
-.4284
-.2705
.70
.2350
-.1925
-.4121
-.3652
.76
.3438
-.0703
-.3501
-.4164
.80
.4600
.0800
-.2330
-.3995
.85
.6838
.2603
-.0506
-.2857
.90
.7160
.4725
.2079
-.0411
.96
.8538
.7184
.6541
.3727
1.00
1.0000
1.0000
1.0000
1.0000
252
TABLE
LEGENDRE POLYNOMIALS P„(cosd)
[Po(cos5) = l]
9
i'i{cos5)
P2(C0Sff)
FgCcosff)
P4(costf)
Psieose)
0°
1.0000
1.0000
1.0000
1.0000
1.0000
5°
.9962
.9886
.9773
.9623
.9437
10°
.9848
.9548
.9106
.8532
.7840
15°
.9659
.8995
.8042
.6847
.5471
20°
.9397
.8245
.6649
.4750
.2715
25°
.9063
.7321
.5016
.2465
.0009
30°
.8660
.6250
.3248
.0234
-.2233
35°
.8192
.5065
.1454
-.1714
-.3691
40°
.7660
.3802
-.0252
-.3190
-.4197
45°
.7071
.2500
-.1768
-.4063
-.3757
50°
.6428
.1198
-.3002
-.4275
-.2545
55°
.5736
-.0065
-.3886
-.3852
-.0868
60°
.5000
-.1250
-.4375
-.2891
.0898
65°
.4226
-.2321
-.4452
-.1552
.2381
70°
.3420
-.3245
-.4130
-.0038
.3281
75°
.2588
-.3995
-.3449
.1434
.3427
80°
.1737
-.4548
-.2474
.2659
.2810
85°
.0872
-.4886
-.1291
.3468
.1577
90°
.0000
-.5000
.0000
.3750
.0000
258
TABLE
43
COMPLETE ELLIPTIC INTEGRALS OF FIRST AND SECOND KINDS
K
, _ - , E = \ yj\ ~- &2 sin^ e d9, fc = sm t&
4-
K
E
0°
1.5708
1.5708
1
1.5709
1.5707
2
1.5713
1.5703
3
1.5719
1.5697
4
1.5727
1.5689
5
1.5738
1.5678
6
1.5751
1.5665
7
1.5767
1.5649
8
1.5785
1.5632
9
1.5805
1.5611
10
1.5828
1.5589
11
1.5854
1.5564
12
1.5882
1.5537
13
1.5913
1.5507
14
1.5946
1.5476
15
1.5981
1.5442
16
1.6020
1.5405
17
1.6061
1.5367
18
1.6105
1.5326
19
1.6151
1.5283
20
1.6200
1.5238
21
1.6252
1.5191
22
1.6307
1.5141
23
1.6365
1.5090
24
1.6426
1.5037
25
1.6490
1.4981
26
1.6557
1.4924
27
1.6627
1.4864
28
1.6701
1.4803
29
1.6777
1.4740
30
1.6858
1.4675
30°
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
K
E
1.6858
1.6941
1.7028
1.7119
1.7214
1.7312
1.7415
1.7522
1.7633
1.7748
1.7868
1.7992
1.8122
1.8256
1.8396
1.8541
1.8691
1.8848
1.9011
1.9180
1.9356
1.9539
1.9729
1.9927
2.0133
2.0347
2.0571
2.0804
2.1047
2.1300
2.1565
1.4675
1.4608
1.4539
1.4469
1.4397
1.4323
1.4248
1.4171
1.4092
1.4013
1.3931
1.3849
1.3765
1.3680
1.3594
1.3506
1.3418
1.3329
1.3238
1.3147
1.3055
1.2963
1.2870
1.2776
1.2681
1.2587
1.2492
1.2397
1.2301
1.2206
1.2111
^
K
E
60°
2.1565
1.2111
61
2.1842
1.2015
62
2.2132
1.1920
63
2.2435
1.1826
64
2.2754
1.1732
65
2.3088
1.1638
66
2.3439
1.1545
67
2.3809
1.1453
68
2.4198
1.1362
m
2.4610
1.1272
70
2.5046
1.1184
71
2.5507
1.1096
72
2.5998
1.1011
73
2.6521
1.0927
74
2.7081
1.0844
75
2.7681
1.0764
76
2.8327
1.0686
77
2.9026
1.0611
78
2.9786
1.0538
79
3.0617
1.0468
80
3.1534
1.0401
81
3.2553
1.0338
82
3.3699
1.0278
83
3.5004
1.0223
84
3.6519
1.0172
85
3.8317
1.0127
86
4.0528
1.0086
87
4.3387
1.0053
88
4.7427
1.0026
89
5.4349
1.0008
90
CO
1.0000
254
TABLE
44
INCOMPLETE ELLIPTIC INTEGRAL OF THE FIRST KIND
J"** dd
y\ — k^ %m^B
0°
10°
20°
30°
40°
50°
60°
70°
80°
90°
0°
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
10^
0.1745
0.1746
0.1746
0.1748
0.1749
0.1751
0.1752
0.1753
0.1754
0.1754
20°
0.3491
0.3493
0.3499
0.3508
0.3520
0.3533
0.3545
0.3565
0.3561
0.3564
30<^
0.5236
0.5243
0.5263
0.5294
0.5334
0.5379
0.5422
0.6459
0.5484
0.6493
40°
0.6981
0.6997
0.7043
0.7116
0.7213
0.7323
0.7436
0.7535
0.7604
0.7629
60°
0.8727
0.8756
0.8842
0.8982
0.9173
0.9401
0.9647
0.9876
1.0044
1.0107
60°
1.0472
1.0519
1.0660
1.0896
1.1226
1.1643
1.2126
1.2619
1.3014
1.3170
70°
1.2217
1.2286
1.2495
1.2853
1.3372
1.4068
1.4944
1.5969
1.6918
1.7354
80°
1.3963
1.4056
1.4344
1.4846
1.5597
1.6660
1.8125
2.0119
2.2653
2.4362
90°
1.5708
1.5828
1.6200
1.6858
1.7868
1.9356
2.1565
2.5046
3.1534
CO
TABLE
45
INCOMPLETE ELLIPTIC INTEGRAL OF THE SECOND KIND
J,*
\/l - fc2 sin^ e dd, k = sin^
\
0°
10°
20°
30°
40°
50°
60°
70°
80°
90°
0°
0.0000
0.0000
0.0000
0.0000
O.OOOO
0.0000
0.0000
0.0000
0.0000
0.0000
10°
0.1745
0.1745
0.1744
0.1743
0.1742
0.1740
0.1739
0.1738
0.1737
0.1736
20°
0.3491
0.3489
0.3483
0.3473
0.3462
0.3450
0.3438
0.3429
0.3422
0.3420
30°
0.5236
0.5229
0.5209
0.6179
0.5141
0.5100
0.5061
0.5029
0.5007
0.5000
40°
0.6981
0.6966
0.6921
0.6851
0.6763
0.6667
0.6575
0.6497
0.6446
0.6428
50°
0.8727
0.8698
0.8614
0.8483
0.8317
0.8134
0.7954
0.7801
0.7697
0.7660
60°
1.0472
1.0426
1.0290
1.0076
0.9801
0.9493
0.9184
0.8914
0.8728
0.8660
70°
1.2217
1.2149
1.1949
1.1632
1.1221
1.0750
1.0266
0.9830
0.9514
0.9397
80°
1.3963
1.3870
1.3697
1.3161
1.2590
1.1926
1.1225
1.0565
1.0054
0.9848
90°
1.5708
1.5589
1.5238
1.4675
1.3931
1.3056
1.2111
1.1184
1.0401
1.0000
255
TABLE
46
ORDINATES OF THE
STANDARD NORMAL CURVE
1
y =
^/%^
-x'/2
X
1
2
3
4
5
6
7
8
9
0.0
.3989
.3989
.3989
.3988
.3986
.3984
.3982
.3980
.3977
.3973
0.1
.3970
.3965
.3961
.3956
.3951
.3945
.3939
.3932
.3925
.3918
0.2
.3910
.3902
.3894
.3885
.3876
.3867
.3857
.3847
.3836
.3825
0.3
.3814
.3802
.3790
.3778
.3765
.3752
.3739
.3725
.3712
.3697
0.4
.3683
.3668
.3653
.3637
.3621
.3605
.3589
.3572
.3555
.3538
0.5
.3521
.3503
.3485
.3467
.3448
.3429
.3410
.3391
.3372
.3352
0.6
.3332
.3312
.3292
.3271
.3251
.3230
.3209
.3187
.3166
.3144
0.7
.3123
.3101
.3079
.3056
.3034
.3011
.2989
.2966
.2943
.2920
0.8
.2897
.2874
.2850
.2827
.2803
.2780
.2756
.2732
.2709
.2685
0.9
.2661
.2637
.2613
.2589
.2565
.2541
.2516
.2492
.2468
.2444
1.0
.2420
.2396
.2371
.2347
.2323
.2299
.2275
.2251
.2227
.2203
1.1
.2179
.2155
.2131
.2107
.2083
.2059
.2036
.2012
.1989
.1965
1.2
.1942
.1919
.1895
.1872
.1849
.1826
.1804
.1781
.1758
.1736
1.3
.1714
.1691
.1669
.1647
.1626
.1604
.1582
.1561
.1539
.1518
1.4
.1497
.1476
.1456
.1435
.1415
.1394
.1374
.1354
.1334
.1315
1.5
.1295
.1276
.1257
.1238
.1219
.1200
.1182
.1163
.1145
.1127
1.6
.1109
.1092
.1074
.1057
.1040
.1023
.1006
.0989
.0973
.0957
1.7
.0940
.0925
.0909
.0893
.0878
.0863
.0848
.0833
.0818
.0804
1.8
.0790
.0775
.0761
.0748
.0734
.0721
.0707
.0694
.0681
.0669
1.9
.0656
.0644
.0632
.0620
.0608
.0596
.0584
.0573
.0562
.0551
2.0
.0540
.0529
.0519
.0508
.0498
.0488
.0478
.0468
.0459
.0449
2.1
.0440
.0431
.0422
.0413
.0404
.0396
.0387
.0379
.0371
.0363
2.2
.0355
.0347
.0339
.0332
.0325
.0317
.0310
.0303
.0297
.0290
2.3
.0283
.0277
.0270
.0264
.0258
.0252
.0246
.0241
.0235
.0229
2.4
.0224
.0219
.0213
.0208
.0203
.0198
.0194
.0189
.0184
.0180
2.5
.0175
.0171
.0167
.0163
.0158
.0154
.0151
.0147
.0143
.0139
2.6
.0136
.0132
.0129
.0126
.0122
.0119
.0116
.0113
.0110
.0107
2.7
.0104
.0101
.0099
.0096
.0093
.0091
.0088
.0086
.0084
.0081
2.8
.0079
.0077
.0075
.0073
.0071
.0069
.0067
.0065
.0063
.0061
2.9
.0060
.0058
.0056
.0055
.0053
.0051
.0050
.0048
.0047
.0046
3.0
.0044
.0043
.0042
.0040
.0039
.0038
.0037
.0036
.0035
.0034
3.1
.0033
.0032
.0031
.0030
.0029
.0028
.0027
.0026
.0025
.0025
3.2
.0024
.0023
.0022
.0022
.0021
.0020
.0020
.0019
.0018
.0018
3.3
.0017
.0017
.0016
.0016
.0015
.0015
.0014
.0014
.0013
.0013
3.4
.0012
.0012
.0012
.0011
.0011
.0010
.0010
.0010
.0009
.0009
3.5
.0009
.0008
.0008
.0008
.0008
.0007
.0007
.0007
.0007
.0006
3.6
.0006
.0006
.0006
.0005
.0005
.0005
.0005
.0005
.0005
.0004
3.7
.0004
.0004
.0004
.0004
.0004
.0004
.0003
.0003
.0003
.0003
3.8
.0003
.0003
.0003
.0003
.0003
.0002
.0002
.0002
.0002
.0002
3.9
.0002
.0002
.0002
.0002
.0002
.0002
.0002
.0002
.0001
.0001
256
TABLE
47
AREAS UNDER THE
STANDARD NORMAL CURVE
from — =o tox
erf(a:) = -^ C e-^^^ dt
X
1
2
3
4
5
6
7
8
9
0.0
.5000
.5040
.5080
.5120
.5160
.5199
.6239
.5279
.5319
.6359
0.1
.5398
.5438
.5478
.5517
.5557
.5596
.5636
.5675
.5714
.5754
0.2
.5793
.5832
.5871
.5910
.5948
.6987
.6026
.6064
.6103
.6141
0.3
.6179
.6217
.6255
.6293
.6331
.6368
.6406
.6443
.6480
.6517
0.4
.6554
.6591
.6628
.6664
.6700
.6736
.6772
.6808
.6844
.6879
0.5
.6915
.6950
.6985
.7019
.7054
.7088
.7123
.7157
.7190
.7224
0.6
.7258
.7291
.7324
.7357
.7389
.7422
.7454
.7486
.7518
.7549
0.7
.7580
.7612
.7642
.7673
.7704
.7734
.7764
.7794
.7823
.7852
0.8
.7881
.7910
.7939
.7967
.7996
.8023
.8051
.8078
.8106
.8133
0.9
.8159
.8186
.8212
.8238
.8264
.8289
.8315
.8340
.8366
.8389
1.0
.8413
.8438
.8461
.8485
.8508
.8531
.8654
.8577
.8699
.8621
1.1
.8643
.8665
.8686
.8708
.8729
.8749
.8770
.8790
.8810
.8830
1.2
.8849
.8869
.8888
.8907
.8925
.8944
.8962
.8980
.8997
.9015
1.3
.9032
.9049
.9066
.9082
.9099
.9115
.9131
.9147
.9162
.9177
1.4
.9192
.9207
.9222
.9236
.9251
.9265
.9279
.9292
.9306
.9319
1.5
.9332
.9345
.9357
.9370
.9382
.9394
.9406
.9418
.9429
,9441
1.6
.9452
.9463
.9474
.9484
.9495
.9506
.9615
.9525
.9535
.9545
1.7
.9554
.9564
.9573
.9582
.9691
.9599
.9608
.9616
.9626
.9633
1.8
.9641
.9649
.9656
.9664
.9671
.9678
.9686
.9693
.9699
.9706
1.9
.9713
.9719
.9726
.9732
.9738
.9744
.9750
.9766
.9761
.9767
2.0
.9772
.9778
.9783
.9788
.9793
.9798
.9803
.9808
.9812
.9817
2.1
.9821
.9826
.9830
.9834
.9838
.9842
.9846
.9860
.9854
.9857
2.2
.9861
.9864
.9868
.9871
.9875
.9878
.9881
.9884
.9887
.9890
2.3
.9893
.9896
.9898
.9901
.9904
.9906
.9909
.9911
.9913
.9916
2.4
.9918
.9920
.9922
.9925
.9927
.9929
.9931
.9932
.9934
.9936
2.5
.9938
.9940
.9941
.9943
.9945
.9946
.9948
.9949
.9951
.9952
2.6
.9953
.9955
.9956
.9957
.9959
.9960
.9961
.9962
.9963
.9964
2.7
.9965
.9966
.9967
.9968
.9969
.9970
.9971
.9972
.9973
.9974
2.8
.9974
.9975
.9976
.9977
.9977
.9978
.9979
.9979
.9980
.9981
2.9
.9981
.9982
.9982
.9983
.9984
.9984
.9985
.9986
.9986
.9986
S.O
.9987
.9987
.9987
.9988
.9988
.9989
.9989
.9989
.9990
.9990
3.1
.9990
.9991
.9991
.9991
.9992
.9992
.9992
.9992
.9993
.9993
3.2
.9993
.9993
.9994
.9994
.9994
.9994
.9994
.9995
.9996
.9995
3.3
.9995
.9995
.9995
.9996
.9996
.9996
.9996
.9996
.9996
.9997
3.4
.9997
.9997
.9997
.9997
.9997
.9997
.9997
.9997
.9997
.9998
3.5
.9998
.9998
.9998
.9998
.9998
.9998
.9998
.9998
.9998
.9998
3.6
.9998
.9998
.9999
.9999
.9999
.9999
.9999
.9999
.9999
.9999
3.7
.9999
.9999
.9999
.9999
.9999
.9999
.9999
.9999
.9999
.9999
3.8
.9999
.9999
.9999
.9999
.9999
.9999
.9999
.9999
.9999
.9999
3.9
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
257
TABLE
48
PERCENTILE VALUES (tp) FOR
STUDENT'S t DISTRIBUTION
with n degrees of freedom
(shaded area = p)
^
\
K
t,>
n
*.995
*.99
*.975
*.95
*.90
^.80
t.7S
^.70
*.60
tss
1
63.66
31.82
12.71
6.31
3.08
1.376
1.000
.727
.325
.158
2
9.92
6.96
4.30
2.92
1.89
1.061
.816
.617
.289
.142
3
5.84
4.54
3.18
2.35
1.64
.978
.765
.584
.277
.137
4
4.60
3.75
2.78
2.13
1.53
.941
.741
.569
.271
.134
5
4.03
3.36
2.57
2.02
1.48
.920
.727
.559
.267
.132
6
3.71
3.14
2.45
1.94
1.44
.906
.718
.553
.265
.131
7
3.50
3.00
2.36
1.90
1.42
.896
.711
.549
.263
.130
8
3.36
2.90
2.31
1.86
1.40
.889
.706
.546
.262
.130
9
3.25
2.82
2.26
1.83
1.38
.883
.703
.543
.261
.129
10
3.17
2.76
2.23
1.81
1.37
.879
.700
.542
.260
.129
11
3.11
2.72
2.20
1.80
1.36
.876
.697
.540
.260
.129
12
3.06
2.68
2.18
1.78
1.36
.873
.695
.539
.259
.128
13
3.01
2.65
2.16
1.77
1.35
.870
.694
.538
.259
.128
14
2.98
2.62
2.14
1.76
1.34
.868
.692
.537
.258
.128
15
2.95
2.60
2.13
1.75
1.34
.866
.691
.536
.258
.128
16
2.92
2.58
2.12
1.75
1.34
.865
.690
.535
.258
.128
17
2.90
2.57
2.11
1.74
1.33
.863
.689
.534
.257
.128
18
2.88
2.55
2.10
1.73
1.33
.862
.688
.534
.257
.127
19
2.86
2.54
2.09
1.73
1.33
.861
.688
.533
.257
.127
20
2.84
2.53
2.09
1.72
1.32
.860
.687
.533
.257
.127
21
2.83
2.52
2.08
1.72
1.32
.859
.686
.532
.257
.127
22
2.82
2.51
2.07
1.72
1.32
.858
.686
.532
.256
.127
23
2.81
2.50
2.07
1.71
1.32
.858
.685
.532
.256
.127
24
2.80
2.49
2.06
1.71
1.32
.857
.685
.531
.256
.127
25
2.79
2.48
2.06
1.71
1.32
.856
.684
.531
.256
.127
26
2.78
2.48
2.06
1.71
1.32
.856
.684
.531
.256
.127
27
2.77
2.47
2.05
1.70
1.31
.855
.684
.531
.256
.127
28
2.76
2.47
2.05
1.70
1.31
.855
.683
.530
.256
.127
29
2.76
2.46
2.04
1.70
1.31
.854
.683
.530
.256
.127
30
2.75
2.46
2.04
1.70
1.31
.854
.683
.530
.256
.127
40
2.70
2.42
2.02
1.68
1.30
.851
.681
.529
.255
.126
60
2.66
2.39
2.00
1.67
1.30
.848
.679
.527
.254
.126
120
2.62
2.36
1.98
1.66
1.29
.845
.677
.526
.254
.126
DO
2.58
2.33
1.96
1.645
1.28
.842
.674
.524
.253
.126
Source: R. A. Fisher and F. Yates, Statistical Tables for Biological, Agricultural and
Medical Research (6th edition, 1963), Table III, Oliver and Boyd Ltd., Edin-
burgh, by permission of the authors and publishers.
258
TABLE
PERCENTILE
VALUES (v^)
FOR
/^-^
*s.
Ar\
THE CH
-SQUARE DISTRIBUTION
/
^s.
\
i
Vr
with n degrees of freedom
/
^^"-^...^
T /
(shaded area
= P)
1
X
2
P
"
n
^.995
^.99
^2
A. 973
^.95
■^.90
Y^
^.75
^.50
^.25
^.05
Y^
A. 025
y2
Y^
1
7.88
6.63
5.02
3.84
2.71
1.32
.455
102
.0158
.0039
.0010
.0002
.0000
2
10.6
9.21
7.38
5.99
4.61
2.77
1.39
575
.211
.103
.0506
.0201
.0100
3
12.8
11.3
9.35
7.81
6.25
4.11
2.37 1.21
.584
.352
.216
.115
.072
4
14.9
13.3
11.1
9.49
7.78
5.39
3.36 1.92
1.06
.711
.484
.297
.207
5
16.7
15.1
12.8
11.1
9.24
6.63
4.35 2.67
1.61
1.15
.831
.554
.412
6
18.5
16.8
14.4
12.6
10.6
7.84
5.35 3.45
2.20
1.64
1.24
.872
.676
7
20.3
18.5
16.0
14.1
12.0
9.04
6.35 4.25
2.83
2.17
1.69
1.24
.989
8
22.0
20.1
17.5
15.5
13.4
10.2
7.34 5.07
3.49
2.73
2.18
1.65
1.34
9
23.6
21.7
19.0
16.9
14.7
11.4
8.34 5.90
4.17
3.33
2.70
2.09
1.73
10
25.2
23.2
20.5
18.3
16.0
12.5
9.34 6.74
4.87
3.94
3.25
2.56
2.16
11
26.8
24.7
21.9
19.7
17.3
13.7
10.3 7.58
6.58
4.57
3.82
3.05
2.60
12
28.3
26.2
23.3
21.0
18.5
14.8
11.3 8.44
6.30
5.23
4.40
3.57
3.07
13
29.8
27.7
24.7
22.4
19.8
16.0
12.3 9.30
7.04
5.89
5.01
4.11
3.57
14
31.3
29.1
26.1
23.7
21.1
17.1
13.3 10.2
7.79
6.57
5.63
4.66
4.07
15
32.8
30.6
27.5
25.0
22.3
18.2
14.3 11.0
8.55
7.26
6.26
5.23
4.60
16
34.3
32.0
28.8
26.3
23.5
19.4
15.3 11.9
9.31
7.96
6.91
5.81
5.14
17
35.7
33.4
30.2
27.6
24.8
20.5
16.3 12.8
10.1
8.67
7.56
6.41
5.70
18
37.2
34.8
31.5
28.9
26.0
21.6
17.3 13.7
10.9
9.39
8.23
7.01
6.26
19
38.6
36.2
32.9
30.1
27.2
22.7
18.3 14.6
11.7
10.1
8.91
7.63
6.84
20
40.0
37.6
34.2
31.4
28.4
23.8
19.3 15.5
12.4
10.9
9.59
8.26
7.43
21
41.4
38.9
35.5
32.7
29.6
24.9
20.3 16.3
13.2
11.6
10.3
8.90
8.03
22
42.8
40.3
36.8
33.9
30.8
26.0
21.3 17.2
14.0
12.3
11.0
9.54
8.64
23
44.2
41.6
38.1
35.2
32.0
27.1
22.3 18.1
14.8
13.1
11.7
10.2
9.26
24
45.6
43.0
39.4
36.4
33.2
28.2
23.3 19.0
15.7
13.8
12.4
10.9
9.89
25
46.9
44.3
40.6
37.7
34.4
29.3
24.3 19.9
16.5
14.6
13.1
11.5
10.5
26
48.3
45.6
41.9
38.9
35.6
30.4
25.3 20.8
17.3
15.4
13.8
12.2
11.2
27
49.6
47.0
43.2
40.1
36.7
31.5
26.3 21.7
18.1
16.2
14.6
12.9
11.8
28
51.0
48.3
44.5
41.3
37.9
32.6
27.3 22.7
18.9
16.9
15.3
13.6
12.5
29
52.3
49.6
45.7
42.6
39.1
33.7
28.3 23.6
19.8
17.7
16.0
14.3
13.1
30
53.7
50.9
47.0
43.8
40.3
34.8
29.3 24.5
20.6
18.5
16.8
15.0
13.8
40
66.8
63.7
59.3
55.8
51.8
45.6
39.3 33.7
29.1
26.5
24.4
22.2
20.7
50
79.5
76.2
71.4
67.5
63.2
56.3
49.3 42.9
37.7
34.8
32.4
29.7
28.0
60
92.0
88.4
83.3
79.1
74.4
67.0
59.3 52.3
46.5
43.2
40.5
37.5
35.5
70
104.2
100.4
95.0
90.5
85.5
77.6
69.3 61.7
55.3
51.7
48.8
45.4
43.3
80
116.3
U2.3
106.6
101.9
96.6
88.1
79.3 71.1
64.3
60.4
57.2
53.5
51.2
90
128.3
124.1
118.1
113.1
107.6
98.6
89.3 80.6
73.3
69.1
65.6
61.8
59.2
100
140.2
135.8
129.6
124.3
118.5
109.1
99.3 90.1
82.4
77.9
74.2
70.1
67.3
Source: Catherine M. Thompson, Table of percentage points of the x^ distribution,
Biometrika, Vol. 32 (1941), by permission of the author and publisher.
259
9Sth PERCENTILE VALUES FOR
THE > DISTRIBUTION
tti = degrees of freedom for numerator
na = degrees of freedom for denominator
(shaded area = .95)
\«1
"A,
1
2
3
4
5
6
8
12
16
20
30
40
50
100
OS
1
161.4
199.5
215.7
224.6
230.2
234.0
238.9
243.9
246.3
248.0
250.1
251.1
252.2
253.0
254.3
2
18.51
19.00
19.16
19.25
19.30
19.33
19.37
19.41
19.43
19.45
19.46
19.46
19.47
19.49
19.50
3
10.13
9.55
9.28
9.12
9.01
8.94
8.86
8.74
8.69
8.66
8.62
8.60
8.58
8.56
8.53
4
7.71
6.94
6.59
6.39
6.26
6.16
6.04
6.91
5.84
5.80
5.75
5.71
5.70
5.66
5.63
5
6.61
5.79
5.41
5.19
5.05
4.95
4.82
4.68
4.60
4.56
4.50
4.46
4.44
4.40
4.36
6
5.99
5.14
4.76
4.53
4.39
4.28
4.16
4.00
3.92
3.87
3.81
3.77
3.75
3.71
3.67
7
5.59
4.74
4.35
4.12
3.97
3.87
3.73
3.57
3.49
3.44
3.38
3.34
3.32
3.28
3.23
8
5.32
4.46
4.07
3.84
3.69
3.58
3.44
3.28
3.20
3.15
3.08
3.05
3.03
2.98
2.93
9
5.12
4.26
3.86
3.63
3.48
3.37
3.23
3.07
2.98
2.93
2.86
2.82
2.80
2.76
2.71
10
4.96
4.10
3.71
3.48
3.33
3.22
3.07
2.91
2.82
2.77
2.70
2.67
2.64
2.59
2.54
11
4.84
3.98
3.59
3.36
3.20
3.09
2.96
2.79
2.70
2.65
2.57
2.53
2.50
2.45
2.40
12
4.75
3.89
3.49
3.26
3.11
3.00
2.85
2.69
2.60
2.54
2.46
2.42
2.40
2.35
2.30
13
4.67
3.81
3.41
3.18
3.03
2.92
2.77
2.60
2.51
2.46
2.38
2.34
2.32
2.26
2.21
14
4.60
3.74
3.34
3.11
2.96
2.85
2.70
2.53
2.44
2.39
2.31
2.27
2.24
2.19
2.13
15
4.64
3.68
3.29
3.06
2.90
2.79
2.64
2.48
2.39
2.33
2.25
2.21
2.18
2.12
2.07
16
4.49
3.63
3.24
3.01
2.85
2.74
2.59
2.42
2.33
2.28
2.20
2.16
2.13
2.07
2.01
17
4.45
3.59
3.20
2.96
2.81
2.70
2.55
2.38
2.29
2.23
2.15
2.11
2.08
2.02
1.96
18
4.41
3.55
3.16
2.93
2.77
2.66
2.51
2.34
2.26
2.19
2.11
2.07
2.04
1.98
1.92
19
4.38
3.52
3.13
2.90
2.74
2.63
2.48
2.31
2.21
2.15
2.07
2.02
2.00
1.94
1.88
20
4.35
3.49
3.10
2.87
2.71
2.60
2.45
2.28
2.18
2.12
2.04
1.99
1.96
1.90
1.84
22
4.30
3.44
3.05
2.82
2.66
2.65
2.40
2.23
2.13
2.07
1.98
1.93
1.91
1.84
1.78
24
4.26
3.40
3.01
2.78
2.62
2.51
2.36
2.18
2.09
2.03
1.94
1.89
1.86
1.80
1.73
26
4.23
3.37
2.98
2.74
2.69
2.47
2.32
2.15
2.06
1.99
1.90
1.85
1.82
1.76
1.69
28
4.20
3.34
2.95
2.71
2.56
2.45
2.29
2.12
2.02
1.96
1.87
1.81
1.78
1.72
1.65
30
4.17
3.32
2.92
2.69
2.53
2.42
2.27
2.09
1.99
1.93
1.84
1.79
1.76
1.69
1.62
40
4.08
3.23
2.84
2.61
2.45
2.34
2.18
2.00
1.90
1.84
1.74
1.69
1.66
1.59
l.Bl
50
4.03
3.18
2.79
2.56
2.40
2.29
2.13
1.96
1.86
1.78
1.69
1.63
1.60
1.62
1.44
60
4.00
3.15
2.76
2.53
2.37
2.25
2.10
1.92
1.81
1.75
1.65
1.59
1.56
1.48
1.39
70
3.98
3.13
2.74
2.50
2.35
2.23
2.07
1.89
1.79
1.72
1.62
1.56
1.63
1.46
1.35
80
3.96
3.11
2.72
2.48
2.33
2.21
2.05
1.88
1.77
1.70
1.60
1.54
1.51
1.42
1.32
100
3.94
3.09
2.70
2.46
2.30
2.19
2.03
1.85
1.75
1.68
1.57
1.51
1.48
1.39
1.28
150
3.91
3.06
2.67
2.43
2.27
2.16
2.00
1.82
1.71
1.64
1.54
1.47
1.44
1.34
1.22
200
3.89
3.04
2.65
2.41
2.26
2.14
1.98
1.80
1.69
1.62
1.52
1.45
1.42
1.32
1.19
400
3.86
3.02
2.62
2.39
2.23
2.12
1.96
1.78
1.67
1.60
1.49
1.42
1.38
1.28
1.13
CO
3.84
2.99
2.60
2.37
2.21
2.09
1.94
1.75
1.64
1.57
1.46
1.40
1.32
1.24
1.00
Source: G. W. Snedecor and W. G. Cochran, Statistical Methods (6th edition, 1967), Iowa
State University Press, Ames, Iowa, by permission of the authors and publisher.
260
99th PERCENTILE VALUES FOR
TABLE
51
THE F DISTRIBUTION
ni = degrees of freedom for numerator
«2 = degrees of freedom for denominator
/"^^
^89
W2\
1
2
3
4
5
6
8
12
16
20
30
40
50
100
CO
1
4052
4999
5403
5625
5764
5859
5981
6106
6169
6208
6258
6286
6302
6334
6366
2
98.49
99.01
99.17
99.25
99.30
99.33
99.36
99.42
99.44
99.45
99.47
99.48
99.48
99.49
99.50
3
34.12
30.81
29.46
28.71
28.24
27.41
27.49
27.05
28.63
26.69
26.50
26.41
26.35
26.23
26.12
4
21.20
18.00
16.69
15.98
15.52
15.21
14.80
14.37
14.15
14.02
13.83
13.74
13.69
13.57
13.46
5
16.26
13.27
12.06
11.39
10.97
10.67
10.27
9.89
9.68
9.65
9.38
9.29
9.24
9.13
9.02
6
13.74
10.92
9.78
9.15
8.75
8.47
8.10
7.72
7.52
7.39
7.23
7.14
7.09
6.99
6.88
7
12.25
9.55
8.45
7.85
7.46
7.19
6.84
6.47
6.27
6.15
5.98
5.90
5.85
5.75
5.66
8
11.26
8.65
7.59
7.01
6.63
6.37
6.03
5.67
5.48
5.36
5.20
5.11
5.06
4.96
4.86
9
10.56
8.02
6.99
6.42
6.06
5.80
5.47
5.11
4.92
4.80
4.64
4.56
4.51
4.41
4.31
10
10.04
7.56
6.55
5.99
5.64
5.39
5.06
4.71
4.52
4.41
4.25
4.17
4.12
4.01
3.91
11
9.05
7.20
6.22
5.67
5.32
5.07
4.74
4.40
4.21
4.10
3.94
3.86
3.80
3.70
3.60
12
9.33
6.93
5.95
5.41
5.06
4.82
4.50
4.16
3.98
3.86
3.70
3.61
3.56
3.46
3.36
13
9.07
6.70
5.74
5.20
4.86
4.62
4.30
3.96
3.78
3.67
3.51
3.42
3.37
3.27
3.16
14
8.86
6.51
5.56
5.03
4.69
4.46
4.14
3.80
3.62
3.51
3.34
3.26
3.21
3.11
3.00
15
8.68
6.36
5.42
4.89
4.66
4.32
4.00
3.67
3.48
3.36
3.20
3.12
3.07
2.97
2.87
16
8.53
6.23
5.29
4.77
4.44
4.20
3.89
3.55
3.37
3.25
3.10
3.01
2.96
2.86
2.75
17
8.40
6.11
5.18
4.67
4.34
4.10
3.79
3.45
3.27
3.16
3.00
2.92
2.86
2.76
2.65
18
8.28
6.01
5.09
4.58
4.25
4.01
3.71
3.37
3.19
3.07
2.91
2.83
2.78
2.68
2.57
19
8.18
6.93
5.01
4.50
4.17
3.94
3.63
3.30
3.12
3.00
2.84
2.76
2.70
2.60
2.49
20
8.10
6.86
4.94
4.43
4.10
3.87
3.56
3.23
3.05
2.94
2.77
2.69
2.63
2.53
2.42
22
7.94
5.72
4.82
4.31
3.99
3.76
3.45
3.12
2.94
2.83
2.67
2.58
2.53
2.42
2.31
24
7.82
5.61
4.72
4.22
3.90
3.67
3.36
3.03
2.85
2.74
2.58
2.49
2.44
2.33
2.21
26
7.72
5.53
4.64
4.14
3.82
3.59
3.29
2.96
2.77
2.66
2.50
2.41
2.36
2.25
2.13
28
7.64
5.45
4.57
4.07
3.76
3.53
3.23
2.90
2.71
2.60
2.44
2.35
2.30
2.18
2.06
30
7.56
5.39
4.51
4.02
3.70
3.47
3.17
2.84
2.66
2.55
2.38
2.29
2.24
2.13
2.01
40
7.31
5.18
4.31
3.83
3.51
3.29
2.99
2.66
2.49
2.37
2.20
2.11
2.05
1.94
1.81
50
7.17
5.06
4.20
3.72
3.41
3.18
2.88
2.56
2.39
2.26
2.10
2.00
1.94
1.82
1.68
60
7.08
4.98
4.13
3.65
3.34
3.12
2.82
2.50
2.32
2.20
2.03
1.93
1.87
1.74
1.60
70
7.01
4.92
4.08
3.60
3.29
3.07
2.77
2.45
2.28
2.15
1.98
1.88
1.82
1.69
1.53
80
6.96
4.88
4.04
3.56
3.25
3.04
2.74
2.41
2.24
2.11
1.94
1.84
1.78
1.65
1.49
100
6.90
4.82
3.98
3.51
3.20
2.99
2.69
2.36
2.19
2.06
1.89
1.79
1.73
1.59
1.43
150
6.81
4.76
3.91
3.44
3.14
2.92
2.62
2.30
2.12
2.00
1.83
1.72
1.66
1.51
1.33
200
6.76
4.71
3.88
3.41
3.11
2.90
2.60
2.28
2.09
1.97
1.79
1.69
1.62
1.48
1.28
400 6.70
4.66
3.83
3.36
3.06
2.85
2.55
2.23
2.04
1.92
1.74
1.64
1.57
1.42
1.19
6.64
4.60
3.78
3.32
3.02
2.80
2.51
2.18
1.99
1.87
1.69
1.59
1.52
1.36
1.00
Source: G. W. Snedecor and W. G. Cochran, Statistical Methods {6th edition, 1967), Iowa
State University Press, Ames, Iowa, by permission of the authors and publisher.
261
9
51772
74640
42331
29044
46621
62898
93582
04186
19640
87056
24033
23491
83587
06568
21960
21387
76105
10863
97453
90581
45939
60173
52078
25424
11645
55870
56974
37428
93507
94271
30586
02133
75797
45406
31041
86707
12973
17169
88116
42187
03585
79353
81938
82322
96799
85659
36081
50884
14070
74950
64937
03355
95863
20790
65304
55189
00745
65253
11822
15804
15630
64759
51135
98527
62586
41889
25439
88036
24034
67283
09448
56301
57683
30277
94623
85418
68829
06652
41982
49159
21631
91157
77331
60710
52290
16835
48653
71590
16159
14676
91097
17480
29414
06829
87843
28195
27279
47152
35683
47280
50532
25496
95652
42457
73547
76552
50020
24819
52984
76168
07136
40876
79971
54195
25708
51817
36732
72484
94923
75936
27989
64728
10744
08396
56242
90985
28868
99431
60995
20507
85184
73949
36601
46253
00477
25234
09908
36574
72139
70186
54398
21154
97810
36764
32869
11785
55261
59009
38714
38723
65544
34371
09591
07839
58892
92843
72828
91341
84821
63886
08263
65952
85762
64236
39238
18776
84303
99247
46149
03229
39817
67906
48236
16057
81812
15815
63700
85915
19219
45943
62257
04077
79443
95203
02479
30763
92486
54083
23631
05825
53298
90276
62545
21944
16530
03878
07516
95715
02526
33537
262
Index of Special Symbols and Notations
The following list shows special symbols and notations used in this book together with pages on which
they are defined or first appear. Cases where a symbol has more than one meaning wiU be clear from
the context.
Symbols
Ber„(a:), Bei„(a!) 140
B{m,n) beta function, 103
B„ Bernoulli numbers, 114
C{x) Fresnel cosine integral, 184
Ci{x) cosine integral, 184
e natural base of logarithms, 1
61,62,63 unit vectors in curvilinear coordinates, 124
erf (x) error function, 183
erfc (x) complementary error function, 183
E = E{k, it/2) complete elliptic integral of second kind, 179
Eik, ,p) incomplete elliptic integral of second kind, 179
EHx) exponential integral, 183
E„ Euler numbers, 114
F(a, b; c; x) hypergeometric function, 160
F{k, 0) incomplete elliptic integral of first kind, 179
y, ^-1 Fourier transform and inverse Fourier transform, 175, 176
^1, fh, Aa scale factors in curvilinear coordinates, 124
Hnix) Hermite polynomials, 151
H^l\x), H'-1\x) Hankel functions of first and second kind, 138
i imaginary unit, 21
i, j,k unit vectors in rectangular coordinates, 117
/„(x) modified Bessel function of first kind, 138
Jf^{x) Bessel function of first kind, 136
K = F{k.Trl2) complete elliptic integral of first kind, 179
Ker„ (a:), Kei„ {x) 140
K^{x) modified Bessel function of second kind, 139
In X or loge X natural logarithm of x, 24
log X OT logio X common logarithm of x, 23
L^{x) Laguerre polynomials, 153
Ln {«) associated Laguerre polynomials, 155
^, ^-1 Laplace transform and inverse Laplace transform, 161
P^{x) Legendre polynomials, 146
^^(2:) associated Legendre functions of first kind, 149
Qnix) Legendre functions of second kind, 148
Q"(x) associated Legendre functions of second kind, 150
r cylindrical coordinate, 49
r polar coordinate, 22, 36
r spherical coordinate, 50
Six) Fresnel sine integral, 184
Si(x) sine integral, 183
r„(a:) Chebyshev polynomials of first kind, 157
U„{x) Chebyshev polynomials of second kind, 158
YAx) Bessel function of second kind, 136
263
264
INDEX OF SPECIAL SYMBOLS AND NOTATIONS
Greek Symbols
y Euler's constant, 1
r{x) gamma function, 1, 101
f(a;) Riemann zeta function, 184
6 cylindrical coordinate, 49
e polar coordinate, 22, 36
e spherical coordinate, 50
«■ 1
•p spherical coordinate, 60
*(p) the sum 1 + - + ^ + ■ ■
'^{x) probability distribution function, 189
+ -, *(0) = 0, 137
Notations
A = B A equals fi or A is equal to B
A > B A is greater than B [or B is less than A]
A < B A is less than B [or B is greater than A]
A ^ £ A is greater than or equal to B
A '^ B A is less than or equal to B
A "^ B A is approximately equal to B
A -^ B A is asymptotic to B or A/B approaches 1, 102
A if A ^
\A\ absolute value of A =
n \ factorial n, 3
'' = g=/».
y =
_ d^y_
dx2
= f"{x), etc.
Dp =
dxP
dy
df sf d-^f
, etc.
dx' dy' dx dy
d{x. y, z)
a(ui,M2,«3)
/
X
f{x) dx
\'dt
A>B
AXB
V
V2 = V • V
V* = V2{V2)
-A if A ^
binomial coefficients, 3
derivatives of y or f{x) witti respect to x, 53, 55
pth derivative with respect to x, 55
differential of y, 55
partial derivatives, 56
Jacobian, 125
indefinite integral, 67
definite integral, 94
line integral of A along C, 121
dot product of A and B, 117
cross product of A and B, 118
del operator, 119
Laplacian operator, 120
biharmonic operator, 120
INDEX
Addition formulas, for Bessel functions, 145
for elliptic functions, 180
for Hermite polynomials, 152
for hyperbolic functions, 27
for trigonometric functions, 15
Agnesi, witch of, 43
Algebraic equations, solutions of, 32, 33
Amplitude, of complex number, 22
of elliptic integral, 179
Analytic geometry, plane [see Plane analytic
geometry]; solid [see Solid analytic geometry]
Angle between lines, in a plane, 35
in space, 47
Annuity, amount of, 201, 242
present value of, 243
Anti-derivative, 57
Antilogarithms, common, 23, 195, 204, 205
natural or Napierian, 24, 226, 227
Archimedes, spiral of, 45
Area integrals, 122
Argand diagram, 22
Arithmetic-geometric series, 107
Arithmetic mean, 185
Arithmetic series, 107
Associated Laguerre polynomials, 155, 156
[see also Laguerre polynomials]
generating function for, 155
orthogonal series for, 156
orthogonality of, 156
recurrence formulas for, 156
special, 155
special results involving, 156
Associated Legendre functions, 149, 150 [see also
Legendre functions]
generating function for, 149
of the first kind, X49
of the second kind, 150
orthogonal series for, 150
orthogonality of, 150
recurrence formulas for, 149
special, 149
Associative law, 117
Asymptotes of hyperbola, 39
Asymptotic expansions or formulas, for Bernoulli
numbers, 115
for Bessel functions, 143
for gamma function, 102
Base of logarithms, 23
change of, 24
Ber and Bei functions, 140, 141
definition of, 140
differential equation for, 141
graphs of, 141
Bernoulli numbers, 98, 107, 114, 116
asymptotic formula for, 115
definition of, 114
relationship to Euler numbers, 115
series involving, 115
table of first few, 114
Bernoulli's differential equation, 104
Bessel functions, 136-145
addition formulas for, 145
asymptotic expansions of, 143
definite integrals involving, 142, 143
generating functions for, 137, 139
graphs of, 141
indefinite integrals involving, 142
infinite products for, 188
integral representations for, 143
modified [see Modified Bessel functions]
of first kind of order n, 136, 137
of order half an odd integer, 138
of second kind of order n, 136, 137
orthogonal series for, 144, 145
recurrence formulas for, 137
tables of, 244-249
zeros of, 250
Bessel's differential equation, 106, 136
general solution of, 106, 137
transformed, 106
Bessel's modified differential equation, 138
general solution of, 139
Beta function, 103
relationship of to gamma function, 103
Biharmonic operator, 120
in curvilinear coordinates, 125
Binomial coefficients, 3
properties of, 4
table of values for, 236, 237
Binomial distribution, 189
Binomial formula, 2
Binomial series, 2, 110
Bipolar coordinates, 128, 129
Laplacian in, 128
Branch, principal, 17
Briggsian logarithms, 23
Cardioid, 41, 42, 44
Cassini, ovals of, 44
Catalan's constant, 181
Catenary, 41
Cauchy or Euler differential equation, 105
Cauchy-Schwarz inequality, 185
for integrals, 186
Cauchy's form of remainder in Taylor series, 110
Chain rule for, derivatives, 53
Characteristic, 194
Chebyshev polynomials, 157-159
generating functions for, 157, 158
of first kind, 157
of second kind, 158
orthogonality of, 158, 159
orthogonal series for, 158, 159
recursion formulas for, 158, 159
relationships involving, 159
special, 157, 158
special values of, 157, 159
Chebyshev's differential equation, 157
general solution of, 169
265
266
INDEX
Chebyshev's inequality, 186
Chi square distribution, 189
percentile values for, 259
Circle, area of, 6
equation of, 37
involute of, 43
perimeter of, 6
sector of [sec Sector of circle]
segment of [see Segment of circle]
Cissoid of Diodes, 45
Common antilogarithms, 23, 195, 204, 205
sample problems involving, 195
table of, 204, 205
Common logarithms, 23, 194, 202, 203
computations using, 196
sample problems involving, 194
table of, 202, 203
Commutative law, for dot products, 118
for vector addition, 117
Complement, 20
Complementary error function, 183
Complex conjugate, 21
Complex inversion formula, 161
Complex numbers, 21, 22, 25
addition of, 21
amplitude of, 22
conjugate, 21
definitions involving, 21
division of, 21, 25
graphs of, 22
imaginary part of, 21
logarithms of, 25
modulus of, 22
multiplication of, 21, 26
polar form of, 22, 25
real part of, 21
roots of, 22, 25
subtraction of, 21
vector representation of, 22
Components of a vector, 117
Component vectors, 117
Compound amount, table of, 240
Cone, elliptic, 51
right circular [see Right circular cone]
Confocal ellipses, 127
ellipsoidal coordinates, 130
hyperbolas, 127
parabolas, 126
paraboloidal coordinates, 130
Conical coordinates, 129
Laplacian in, 129
Conies, 37 [see also Ellipse, Parabola, Hyperbola]
Conjugate, complex, 21
Constant of integration, 57
Convergence, interval of, 110
of Fourier series, 131
Convergence factors, table of, 192
Coordinate curves, 124
system, 11
Coordinates, curvilinear, 124-130
cylindrical, 49, 126
polar, 22, 36
rectangular, 36, 117
Coordinates, curvilinear (cont.)
rotation of, 36, 49
special orthogonal, 126-130
spherical, 50, 126
transformation of, 36, 48, 49
translation of, 36, 49
Cosine integral, 184
Fresnel, 184
table of values for, 251
Cosines, law of for plane triangles, 19
law of for spherical triangles, 19
Counterclockwise, 11
Cross or vector product, 118
Cube, duplication of, 45
Cube roots, table of, 238, 239
Cubes, table of, 238, 239
Cubic equation, solution of, 32
Curl, 120
in curvilinear coordinates, 125
Curtate cycloid, 42
Curves, coordinate, 124
special plane, 40-45
Curvilinear coordinates, 124, 125
orthogonal, 124-130
Cycloid, 40, 42
curtate, 42
prolate, 42
Cylinder, elliptic, 51
lateral surface area of, 8, 9
volume of, 8, 9
Cylindrical coordinates, 49, 126
Laplacian in, 126
Definite integrals, 94-100
approximate formulas for, 95
definition of, 94
general formulas involving, 94, 95
table of, 95-100
Degrees, 1, 199, 200
conversion of to radians, 199, 200, 223
relationship of to radians, 12, 199, 200
Del operator, 119
miscellaneous formulas involving, 120
Delta function, 170
DeMoivre's theorem, 22, 25
Derivatives, 53-56 [see also Differentiation]
anti-, 57
chain rule for, 53
definition of, 53
higher, 55
of elliptic functions, 181
of exponential and logarithmic functions, 54
of hyperbolic and inverse hyperbolic
functions, 54, 55
of trigonometric and inverse trigonometric
functions, 64
of vectors, 119
partial, 56
Descartes, folium of, 43
Differential equations, solutions of basic, 104-106
Differentials, 55
rules for, 56
Differentiation, 53 [see also Derivatives]
INDEX
267
Differentiation (cont.)
general rules for, 63
of integrals, 95
Diodes, cissoid of, 45
Direction cosines, 46, 47
numbers, 46, 48
Directrix, 37
Discriminant, 32
Distance, between two points in a plane, 34
between two points in space, 46
from a point to a line, 35
from a point to a plane, 48
Distributions, probability, 189
Distributive law, 117
for dot products. 118
Divergence, 119
in curvilinear coordinates, 125
Divergence theorem, 123
Dot or scalar product, 117, 118
Double angle formulas, for hyperbolic functions, 27
for trigonometric functions, 16
Double integrals, 122
Duplication formula for gamma functions, 102
Duplication of cube, 45
Eccentricity, definition of, 37
of ellipse, 38
of hyperbola, 39
of parabola, 37
Ellipse. 7, 37, 38
area of, 7
eccentricity of, 38
equation of, 37, 38
evolute of, 44
focus of, 38
perimeter of, 7
semi-major and-minor axes of, 7, 38
Ellipses, confocal, 127
Ellipsoid, equation of, 51
volume of, 10
Elliptic cone, 51
cylinder, 51
paraboloid, 52
Elliptic cylindrical coordinates, 127
Laplacian in, 127
Elliptic functions, 179-182 [see also Elliptic
integrals]
addition formulas for, 180
derivatives of, 181
identities involving, 181
integrals of, 182
Jacobi's, 180
periods of, 181
series expansions for, 181
special values of, 182
Elliptic integrals, 179, 180 [see also Elliptic functions]
amplitude of, 179
Landen's transformation for, 180
Legendre's relation for, 182
of the first kind, 179
of the second kind, 179
of the third kind, 179, 180
table of values for, 254, 255
Envelope, 44
Epicycloid, 42
Equation of line, 34
general, 35
in parametric form, 47
in standard form, 47
intercept form for, 34
normal form for, 35
perpendicular to plane, 48
Equation of plane, general, 47
intercept form for, 47
normal form for, 48
passing through three points, 47 —
Error function, 183
complementary, 183
table of values of, 257
Euler numbers, 114, 115
definition of, 114
relationship of, to Bernoulli numbers, 115
series involving, 115
table of first few, 114
Euler or Cauchy differential equation, 105
Euler-Maclaurin summation formula, 109
Euler's constant, 1
Euler's identities, 24
Evolute of an ellipse, 44
Exact differential equation, 104
Exponential functions, 23-25, 200
periodicity of, 24
relationship of to trigonometric functions, 24
sample problems involving calculation of, 200
series for. 111
table of, 226, 227
Exponential integral, 183
table of values for, 261
Exponents, 23
F distribution, 189
95th and 99th percentile values for, 260, 261
Factorial n, 3
table of values for, 234
Factors, 2
Focus, of conic, 37
of ellipse, 38
of hyperbola, 39
of parabola, 38
Folium of Descartes, 43
Fourier series, 131-135
complex form of, 131
convergence of, 131
definition of, 131
Parseval's identity for, 131
special, 132-135
Fourier transforms, 174-178 _ •
convolution theorem for, 175
cosine, 176
definition of, 175
Parseval's identity for, 175
sine, 175
table of, 176-178
Fourier's integral theorem, 174
Fresnel sine and cosine integrals, 184
268
INDEX
Frullani's integral, 100
Frustrum of right circular cone, lateral surface
area of, 9
volume of, 9
Gamma function, 1, 101, 102
asymptotic expansions for, 102
definition of, 101, 102
derivatives of, 102
duplication formula for, 102
for negative values, 101
graph of, 101
infinite product for, 102, 188
recursion formula for, 101
relationship of to beta function, 103
relationships involving, 102
special values for, 101
table of values for, 235
Gaussian plane, 22
Gauss' theorem, 123
Generalized integration by parts, 59
Generating functions, 137, 139, 146, 149, 161, 153,
155, 157, 158
Geometric formulas, 6-10
Geometric mean, 185
Geometric series, 107
arithmetic-, 107
Gradient, 119
in curvilinear coordinates, 125
Green's first and second identities, 124
Green's theorem, 123
Half angle formulas, for hyperbolic functions, 27
for trigonometric functions, 16
Half rectified sine wave function, 172
Hankel functions, 138
Harmonic mean, 185
Heaviside's unit function, 173
Hermite polynomials, 151, 162
addition formulas for, 152
generating function for, 151
orthogonal series for, 152
orthogonality of, 152
recurrence' formulas for, 151
Rodrigue's formula for, 151
special, 151
special results involving, 152
Hermite's differential equation, 151
Higher derivatives, 55
Leibnitz rule for, 55
Holder's inequality, 185
for integrals, 186
Homogeneous differential equation, 104
linear second order, 105
Hyperbola, 37, 39
asymptotes of, 39
eccentricity of, 39
equation of, 37
focus of, 39
length of major and minor axes of, 39
Hyperbolas, confocal, 127
Hyperbolic functions, 26-31
addition formulas for, 27
Hyperbolic functions (cont.)
definition of, 26
double angle formulas for, 27
graphs of, 29
half angle formulas for, 27
inverse [see Inverse hyperbolic functions]
multiple angle formulas for, 27
of negative arguments, 26
periodicity of, 31
powers of, 28
relationship of to trigonometric functions, 31
relationships among, 26, 28
sample problems for calculation of, 200, 201
series for, 112
sum, difference and product of, 28
table of values for, 228-233
Hyperbolic paraboloid, 52
Hyperboloid, of one sheet, 51
of two sheets, 52
Hypergeometric differential equation, 160
distribution, 189
Hypergeometric functions, 160
miscellaneous properties of, 160
special cases of, 160
Hypocycloid, general, 42
with four cusps, 40
Imaginary part of a complex number, 21
Imaginary unit, 21
Improper integrals, 94
Indefinite integrals, 57-93
definition of, 57
table of, 60-93
transformation of, 59, 60
Inequalities, 185, 186
Infinite products, 102, 188
series [see Series]
Initial point of a vector, 116
Integral calculus, fundamental theorem of, 94
Integrals, definite [see Definite integrals]
double, 122
improper, 94
indefinite [see Indefinite integrals]
involving vectors, 121
line [see Line integrals]
multiple, 122, 125
Integration, 67 [see also Integrals]
constants of, 57
general rules of, 57-59
Integration by parts, 57
generalized, 69
Intercepts, 34, 47
Interest, 201, 240-243
Interpolation, 195
Interval of convergence, 110
Inverse hyperbolic functions, 29-31
definition of, 29
expressed in terms of logarithmic functions, 29
graphs of, 30
principal values for, 29
relationship of to inverse trigonometric
functions, 31
relationships between, 30
INDEX
269
Inverse Laplace transforms, 161
Inverse trigonometric functions, 17-19
definition of, 17
graphs of, 18, 19
principal values for, 17
relations between, 18
relationship of to inverse hyperbolic
functions, 31
Involute of a circle, 43
Jacobian, 125
Jacobi's elliptic functions, 180
Ker and Kei functions, 140, 141
definition of, 140
differential equation for, 141
£:raphs of, 141
Lagrange form of remainder in Taylor series, 110
Laguerre polynomials, 153, 154
associated [see Associated Laguerre polynomials]
generating function for, 153
orthogonal series for, 164
orthogonality of, 154
recurrence formulas for, 163
Rodrigue's formula for, 153
special, 153
Laguerre's associated differential equation, 155
Laguerre's differential equation, 153
Landen's transformation, 180
Laplace transforms, 161-173
complex inversion formula for, 161
definition of, 161
inverse, 161
table of, 162-173
Laplacian, 120
in curvilinear coordinates, 125
Legendre functions, 146-148 [see also Legendre
polynomials]
associated [see Associated Legendre functions]
of the second kind, 148
Legendre polynomials, 146, 147 [see also
Legendre functions]
generating function for, 146
orthogonal series of, 147
orthogonality of, 147
recurrence formulas for, 147
Rodrigue's formula for, 146
special, 146
special results involving, 147
table of values for, 252, 253
Legendre's associated differential equation, 149
general solution of, 150
Legendre's differential equation, 106, 146
general solution of, 148
Legendre's relation for elliptic integrals, 182
Leibnitz's rule, for differentiation of integrals, 95
for higher derivatives of products, 55
Lemniscate, 40, 44
Limacon of Pascal, 41, 44
Line, equation of [see Equation of line]
integrals [see Line integrals]
slope of, 34
Linear first order differential equation, 104
second order differential equation, 105
Line integrals, 121, 122
definition of, 121
independence of path of, 121, 122
properties of, 121
Logarithmic functions, 23-25 [see also Logarithms]
series for. 111
Logarithms, 23 [see also Logarithmic functions]
antilogarithms and [see Antilogarithms]
base of, 23
Briggsian, 23
change of base of, 24
characteristic of, 194
common [see Common logarithms]
mantissa of, 194
natural, 24
of complex numbers, 25
of trigonometric functions, 216-221
Maclaurin series, 110
Mantissa, 194
Mean value theorem, for definite integrals, 94
generalized, 95
Minkowski's inequality, 186
for integrals, 186
Modified Bessel functions, 138, 139
differential equation for, 138
generating function for, 139
graphs of, 141
of order half an odd integer, 140
recurrence formulas for, 139
Modulus, of a complex number, 22
Moments of inertia, special, 190, 191
Multinomial formula, 4
Multiple angle formulas, for hyperbolic
functions, 27
for trigonometric functions, 16
Multiple integrals, 122
transformation of, 125
Napierian logarithms, 24, 196
tables of, 224, 225
Napier's rules, 20
Natural logarithms and antilogarithms, 24, 196
tables of, 224-227
Neumann's function, 136
Nonhomogeneous equation, linear second order, 105
Normal, outward drawn or positive, 123
unit, 122
Normal curve, areas under, 257
ordinates of, 256
Normal distribution, 189
Normal form, equation of line in, 35
equation of plane in, 48
Null function, 170
Null vector, 116
Numbers, complex [see Complex numbers]
Oblate spheroidal coordinates, 128 ,
Laplacian in, 128
Orthogonal curvilinear coordinates, 124-130
formulas involving, 125
270
INDEX
Orthogonality and orthogonal aeries, 144, 145,
147, 150, 152, 154, 156, 168, 159
Ovals of Cassini, 44
Parabola, 37, 38
eccentricity of, 37
equation of, 37, 38
focus of, 38
segment of [see Segment of parabola]
Parabolas, confocal, 126
Parabolic cylindrical coordinates, 126
Laplacian in, 126
Parabolic formula for definite integrals, 95
Paraboloid elliptic, 62
hyperbolic, 52
Paraboloid of revolution, volume of, 10
Paraboloidal coordinates, 127
Laplacian in, 127
Parallel, condition for lines to be, 35
Parallelepiped, rectangular [see Rectangular
parallelepiped]
volume of, 8
Parallelogram, area of, 5
perimeter of, 5
Parallelogram law for vector addition, 116
Parseval's identity, for Fourier transforms, 175
for Fourier series, 131
Partial derivatives, 56
Partial fraction expansions, 187
Pascal, limacon of, 41, 44
Pascal's triangle, 4, 236
Perpendicular, condition for lines to be, 35
Plane, equation of [see Equation of plane]
Plane analytic geometry, formulas from, 34-39
Plane triangle, area of, 5, 36
law of cosines for, 19
law of sines for, 19
law of tangents for, 19
perimeter of, 5
radius of circle circumscribing, 6
radius of circle inscribed in, 6
relationships between sides and angles of, 19
Poisson distribution, 189
Poisson summation formula, 109
Polar coordinates, 22, 36
transformation from rectangular to, 36
Polar form, expressed as an exponential, 25
multiplication and division in, 22
of a complex number, 22, 25
operations in, 25
Polygon, regular [see Regular polygon]
Power, 23
Power series, 110
reversion of, 113
Present value, of an amount, 241
of an annuity, 243
Principal branch, 17
Principal values, for inverse hyperbolic functions, 29
for inverse trigonometric functions, 17, 18
Probability distributions, 189
Products, infinite, 102, 188
special, 2
Prolate cycloid, 42
Prolate spheroidal coordinates, 128
Laplacian in, 128
Pulse function, 173
Pyramid, volume of, 9
Quadrants, 11
Quadratic equation, solution of, 32
Quartic equation, solution of, 33
Radians, 1, 12, 199, 200
relationship of to degrees, 12, 199, 200
table for conversion of, 222
Random numbers, table of, 262
Real part of a complex number, 21
Reciprocals, table of, 238, 239
Rectangle, area of, 5
perimeter of, 5
Rectangular coordinate system, 117
Rectangular coordinates, transformation of to
polar coordinates, 36
Rectangular formula for definite integrals, 95
Rectangular parallelepiped, volume of, 8
surface area of, 8
Rectified sine wave function, 172
half, 172
Recurrence or recursion formulas, 101, 137, 139,
147, 149, 151, 153, 156, 158, 159
Regular polygon, area of, 6
circumscribing a circle, 7
inscribed in a circle, 7
perimeter of, 6
Reversion of power series, 113
Riemann zeta function, 184
Right circular cone, frustrum of
[see Frustrum of right circular cone]
lateral surface area of, 9
volume of, 9
Right-handed system, 118
Rodrigue's formulas, 146, 151, 153
Roots, of complex numbers, 22, 25
liable of square and cube, 238, 239
Rose, three- and four-leaved, 41
Rotation of coordinates, in a plane, 36
in space, 49
Saw tooth wave function, 172
Scalar or dot product, 117, 118
Scalars, 116
Scale factors, 124
Schwarz inequality [see Cauchy-Schwarz inequality]
Sector of circle, arc length of, 6
area of, 6
Segment of circle, area of, 7
Segment of parabola, area of, 7
arc length of, 7
Separation of variables, 104
Series, arithmetic, 107
arithmetic-geometric, 107
binomial, 2, 110
Fourier [see Fourier series]
geometric, 107
of powers of positive integers, 107, 108
of reciprocals of powers of positive integers,
108, 109
INDEX
271
Series, arithmetic (cont.)
orthogonal [see Orthogonality and orthogonal series]
power, HO, 113
Taylor [see Taylor series]
Simple closed curve, 123
Simpson's formula for definite integrals, 95
Sine integral, 183
Fresnel, 184
table of values for, 251
Sines, law of for plane triangle, 19
law of for spherical triangle, 19
Slope of line, 34
Solid analytic geometry, formulas from, 46-52
Solutions of algebraic equations, 32, 33
Sphere, equation of, 50
surface area of, 8
triangle on [see Spherical triangle]
volume of, 8
Spherical cap, surface area of, 9
volume of, 9
Spherical coordinates, 50, 126
Laplacian in, 126
Spherical triangle, area of, 10
Napier's rules for right angled, 20
relationships between sides and angles of, 19, 20
Spiral of Archimedes, 45
Square roots, table of, 238, 239
Square wave function, 172
Squares, table of, 238, 239
Step function, 173
Stirling's asymptotic series, 102
formula, 102
Stoke's theorem, 123
Student's t distribution, 189
percentile values for, 258
Summation formula, Euler-Maclaurin, 109
Poisson, 109
Sums [see Series]
Surface integrals, 122
relation of to double integral, 123
Tangent vectors to curves, 124
Tangents, law of for plane triangle, 19
law of for spherical triangle, 20
Taylor series, 110-113
for functions of one variable, 110
for functions of two variables, 113
Terminal point of a vector, 116
Toroidal coordinates, 129
Laplacian in, 129
Torus, surface area of, 10
volume of, 10
Tractrix, 43
Transformation, Jacobian of, 125
of coordinates, 36, 48, 49, 124
of integrals, 59, 60, 125
Translation of coordinates, in a plane, 36
in space, 49
Trapezoid, area of, 5
perimeter of, 5
Trapezoidal formula for definite integrals, 96
Triangle, plane [see Plane triangie]
spherical [see Spherical triangle]
Triangle inequality, 185
Triangular wave function, 172
Trigonometric functions, 11-20
addition formxilas for, 15
definition of, 11
double angle formulas for, 16
exact values of for various angles, 13
for various quadrants in terms of
quadrant I, 15
general formulas involving, 17
graphs of, 14
half angle formulas, 16
inverse [see Inverse trigonometric functions]
multiple angle formulas for, 16
of negative angles, 14
powers of, 16
relationship of to exponential functions, 24
relationship of to hyperbolic functions, 31
relationships among, 12, 15
sample problems involving, 197-199
series for, 111
signs and variations of, 12
sum, difference and product of, 17
table of in degrees and minutes, 206-211
table of in radians, 212-215
table of logarithms of, 216-221
Triple integrals, 122
Trochoid, 42
Unit function, Heaviside's, 173
Unit normal to a surface, 122
Unit vectors, 117
Vector algebra, laws of, 117
Vector analysis, formulas from, 116-130
Vector or cross product, 118
Vectors, 116
addition of, 116, 117
complex numbers as, 22
components of, 117
equality of, 117
fundamental definitions involving, 116, 117
multiplication of by scalars, 117
notation for, 116
null, 116
parallelogram law for, 116
sums of, 116, 117
tangent, 124
unit, 117
Volume integrals, 122
Wallis' product, 188
Weber's function, 136
Witch of Agnesi, 43
X axis, 11
z intercept, 34
y axis, 11
y intercept, 34
Zero vector, 116
Zeros of Bessel functions, 250
Zeta function of Riemann, 184
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